an introduction to the wiener-hopf technique for the...
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8/29/2007 D:\Users\vdaniele\libri\WHnew\preface.doc
AN INTRODUCTION TO THE WIENER-HOPF TECHNIQUE FOR THE SOLUTION
OF ELECTROMAGNETIC PROBLEMS Introduction In 1931 Wiener and Hopf [1] invented a powerful technique for solving an integral equation of a special type. By introducing the Laplace transform of the unknown, the integral equation has been rephrased in terms of a functional equation defined in a suitably defined complex space. The solution method of the latter is very ingenious indeed. It is based on a sophisticated procedure exploiting some properties of the analytic functions and, to this author’s opinion, it stands as one of the most important mathematical inventions. In fact an enormous variety of physically relevant problems can be expressed in terms of the equation originally solved by Wiener and Hopf or to strictly connected ones. In diffraction problems a fundamental approach due to Jones [2] applies the Laplace transforms directly to the partial differential equations, and the complex variable functional equations are so obtained without having to formulate an integral equation before. The Jones approach has been adopted systematically by Noble [3] in his fundamental book on the Wiener-Hopf technique. The book by Noble presents many applications of the Wiener-Hopf technique in a systematic way and it is a fundamental book for the readers interested in this powerful method. Unfortunately this book has been written many years ago, and in the meantime the Wiener Hopf technique has been the object of the scientific attention of many scientists and important developments have been achieved. The main aim of this book is to provide now, first of all, for the diffraction students and scientists a comprehensive introduction to the Wiener-Hopf technique including its latest up to date developments obtained by this author. In particular, these developments showed the wide range of possible applications of this technique. In practice it is possible to say that nowadays all the canonical diffraction problems involving geometrical discontinuities can be successfully tackled by the Wiener-Hopf technique, and this fact has definitively established this technique as the most general and powerful analytical method for solving diffraction field problems.
The number of problems that can be effectively approached by the W-H technique is very high, as it is possible to see even in fig.0.1 where some of them are depicted. In few words, all these are geometrical structures that can be considered as equivalent to a (uniform or non-uniform) waveguide where semi-infinite geometrical discontinuities have been introduced. These discontinuities may be also modified in the transversal or longitudinal direction of the wave guide, and this considerably increases the number of possible problems that can be effectively studied by this technique. It must be observed too that most of these problems are very important problems, and that often there are no alternative approaches available for solving them, even numerically. However, some remarks about W-H techniques are necessary before going deep through them. First of all, no W-H problem is simple to study. For instance, for a given electromagnetic problem that perhaps may be formulated in terms of W-H equations, it could be even a very difficult problem to obtain these equations. In particular the key function that contain all the information of the studied problem is the W-H kernel that generally is a matrix G function of a complex variable α . To find ( )αG for a specific geometry requires a deep knowledge both of the formulation of electromagnetic problems and of the electromagnetic physical concepts. Furthermore, other great difficulties must be faced for solving the functional equations, even though their solutions arise from a conceptually very simple problem: the factorization of the matrix ( )αG . This problem constitutes also a very beautiful mathematical problem that in the past has become a ‘cult’ activity for many students. Finally new efforts are necessary to work out fruitful solutions too: in fact, even if formal solutions may be obtained, a long and difficult elaboration is always required to make them effective from the physical and the engineering points of view.
fig.0.1: Some few examples of W-H geometries
Concerning the overall philosophy of the subject presentation, this book has been written for readers primarily interested in the ideas and in the possible applications of the presented methods. For this reason, the considered arguments are often only delineated and not pursued on nor discussed in extreme mathematical depth. For what concerns the adopted notations, we have to say that throughout this book we consider only time harmonic fields with a time dependence specified by the factor tje ω (electrical engineering notations), which is omitted, and where the imaginary unity is indicated with j. Conversely, in applied mathematics the factor tje ω becomes tie ω− . It means that in the natural domain the change ij −⇒ transforms engineering notations in applied mathematics notations (and vice versa). However, in the spectral domain usually the same notations are used in engineering and applied mathematics. In fact, regarding for example the Fourier transforms, the following definitions are the most frequently used in literature:
∫∞
∞−= dxexfF xj
eeαα )()( , dxexfF xi
aa ∫∞
∞−= αα )()(
where the subscript e means engineering and the subscript a means applied mathematics. Consequently in the spectral domain on the real axis we have:
)()( αα −= ea FF and j substituted by –i (and vice versa) For example, let us consider in the natural domain the propagation factor that is defined in electrical engineering notations by:
xkje exf −=)( with the propagation constant k defined by:
jak −= β , 0≥a
the same propagation factor in applied mathematics notations is written:
xkia
aexf =)( with αβ ika += In the Laplace domain, on the real axis, we have:
kjdxeexfF xjxkj
ee −== ∫
∞ −
αα α
0)()(
that in applied mathematics notations is written:
( ) ( )e aa a
j i iF Fk k k
α αα α α
−= ⇒ = =
− − − +
Analytic continuations define the previous functions in the whole complex plane α . It means that the Laplace Transform are defined for every value of α by :
kjFe −
=α
α )( . a
a kiF+
=α
α )(
In the following we will define plus ( )F α+ and minus ( )F α− (see 1.1.1). Notice that a plus (or minus) function (1.1.1) in the electrical engineering notations is also a plus (or minus) function in the applied mathematics notations. The only difference is given by the position of the singularities. In fact, for example ( ) and ( )e aF Fα α defined before are plus functions both with “engineering” and “applied mathematics” notation. However
kjFe −
=α
α )( has a singularity in jak −== βα and a
a kiF+
=α
α )( in
ak iaα β= − = − − .
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8/29/2007 D:\Users\vdaniele\libri\WHnew\chapter 1-1.doc 1.1. Wiener-Hopf (W-H) equations forms 1.1.1. The Basic W-H Equation W-H equations may be considered a generalization of the well known integral convolution equation:
(1) )(' )'()'( xfdxxfxxg o=−∫∞
∞−, ∞<<∞− x
where )(xf is the unknown function. In the Fourier transformed domain this equation may be solved by the formula: (2) )()()( 1 ααα oFGF −= where )(αF , )(αG and )(αoF are the Fourier transforms of, respectively, )(xf , )(xg and )(xfo , according to the definition
∫∞
∞−= dxexfF xjαα )()( ,
∫∞
∞−= dxexgG xjαα )()( ,
∫∞
∞−= dxexfF xj
ooαα )()( .
The W-H integral equation consists in assuming the integration on a semi-infinite domain
∞−0 :
(3) )(' )'()'(0
xfdxxfxxg o=−∫∞
, ∞<< x0
For the presence of the semi-infinite domain of definition the Wiener-Hopf equation is considerably difficult to tackle, and it was only in the fundamental work by Wiener and Hopf [1] that the explicit solutions were obtained for the very first time. In order to effectively grasp the difference between the W-H equation (3) and the convolution equation (1) it is very useful to rewrite eq. (3) as a convolution product:
(4) ))(1)(()( )(')'( )'()'( xuxfxuxfdxxuxfxxg o −+=− −
∞
∞−∫ , ∞<<∞− x
2
where )(xu is the unit step function and )(xf − a new unknown representing the continuation on the left hand side of eq.(3) for 0<x . By Fourier transforming eq. (4), it yields the following functional equation: (5) )()()()( αααα +−+ += oFFFG where )(α+oF and )(α−F are, respectively, the Fourier transforms of the right axis function )()( xuxfo and of the left-axis function ))(1)(( xuxf −− . From here onwards, we will refer to the W-H equations as stated here in (5), in their spectral form. The W-H equations can be classified as scalar or vector, the former involves only scalar quantities, the latter also vector quantities of order n, such as )(α+F , )(o α+F and )(α−F , and matrix quantities, such as )(αG , a square matrix of the same order. The scalar case, the simplest class of W-H equations, can always be solved in closed form [1]; conversely, for the vector case, there is no closed-form solution for the general case. In fact, in spite of the great efforts by many scientists, there has been little progress towards a general method of explicit solution in this case. We can say that the state of the art is now that in the seventy years since the solution of the scalar equation was found, the vector equation has been solved in closed forms only in very particular cases, for specific forms of the matrix )(αG . However, efficient approximate solution procedures to handle effectively the problem in these cases apply, and these will be described in this book. In the W-H equations the unknowns involved are the functions )(α+F and )(α−F , being
)(αG and )(α+oF known functions. )(α+F and )(α−F are unilateral, one-sided, Laplace transforms so defined:
α
α
αααjs
js
zszjzj zfLdzezfdzezfdzezuzfF−=
−=
∞ −∞∞
∞−+ ==== ∫∫∫ )([)()()()()(00
αα
αααjs
js
zszjzj zfLdzezfdzezfdzezuzfF=−
=
∞
−∞− −
∞
∞− −− −=−==−= ∫∫∫ )([)()())(1)(()(0
0
In the following )(α+F and ( )F α− will be called respectively plus and minus function. Taking into account the properties of the unilateral Laplace Transforms, )(α+F is infinite regular (in the following regular or analytic) in an upper half-plane of the complex plane where it is vanishing as ∞→α , while conversely )(α−F is regular in a lower half-plane where it is vanishing as ∞→α . We are dealing with the so defined conventional plus (and minus) functions, if they are regular in the half-planes 0]Im[ ≥α and 0]Im[ ≤α respectively.
3
Besides the classical Wiener-Hopf equations (CWHE), previously defined by eq. (5), there are many electromagnetic field problems leading to functional equations that we can classify as modified W-H equations, generalized W-H equations and modified generalized W-H equations, as it will be defined in the following. Even if it is possible to show in many cases the equivalence of these functional equations to the classical W-H equations (5), sometimes it turns out to be more convenient to solve them with particular “ad hoc“ dedicated approaches. Let us start to consider in the next chapter these other classes of W-H equations, their properties and how to handle them. Before discussing the generalizations of the W-H equations we remind that in literature other forms of the W-H equations are known. In particular, it is important the formulation based on the two following dual integral equations:
(6a) )()()( xfdeFG oxj =∫
∞
∞−
−+ ααα α , for 0x ≥
(6b) 0)( =∫∞
∞−
−− αα α deF xj , for 0<x
Equations (6) are called after Karp-Weinstein-Clemmow and can be easily obtained by inverse transforming equation (5) and splitting it into two equations, related to the plus and minus functions. 1.1.2. Modified W-H equations (MWHE)
1.1.2.1. Longitudinally modified W-H equations If in the equation (1) the convolution integral has been defined on a finite support
Lx << '0 , the integral equation so stands:
(7) ∫ ≤≤=−L
o Lxxfdxxfxxg0
0)(')'()'(
By extending this equation to all the real axis +∞<<∞− x , it yields:
)()()(')'()'( 21 xfxfxfdxxfxxg o∫∞
∞−++=− , +∞<<∞− x
where the known function )(xfo has finite support on the original integration support interval ],0[ L and the three unknown )(xf , )(1 xf and )(2 xf are respectively defined on
],0[ L , ( 0,∞− ], [ +∞,L ). Introducing the Fourier transform of the last equation it yields: (8) )(F)(F)(Fe)(F)(G o
Lj α=α+α+αα −+α
where:
4
)(G α and )(Fo α are Fourier transforms of, respectively, )(xg and )(xfo , the latter
with compact support ],0[ L ; similarly, dxexfFL xj∫= 0
)()( αα is the continuous Fourier
transform of the portion of )(xf defined in the same compact support ],0[ L ; it can be shown that the entire function )(αF has algebraic behavior as +∞→]Im[α . On the other hand, we define the minus and plus functions )(F α− and )(F α+ as opposite of the Fourier transforms of )(1 xf and the shifted )Lx(f 2 +
dxexfF xj∫ ∞−− −=0
1 )()( αα
x~de)Lx~(fdxe)x(fe)(F0
x~j2L
xj2
Lj ∫∫∞ α∞ αα−
+ +−=−=α
We call eq.(8) longitudinally modified W-H equation, and this equation is present in many areas of applied mathematics, i.e. in diffraction problems it arises from geometries like slit-coupled waveguides or truncated waveguides where longitudinal modifications have been assumed in a uniform or non-uniform guiding structure, but it can also arise in other completely different applied mathematics areas, such as Mathematical Finance. It is worth noticing that the modified W-H equation can be reduced to a classical vector W-H equation of order two. (see section 1.2.11)
1.1.2.2. Transversely modified W-H equations. The transversely modified W-H equations are so defined: (9) )()()()()()( 0 αααααα +−++ =+−+ FFFHFG By comparing eq. (9) and eq. (5), it stands that the transversely modified W-H equations are characterized by the presence of the additional term )()( αα −+FH that does not appear in the previous one. The adverb ‘transversely’ is due to the fact that this equation appears when the semi-infinite discontinuity introduced in the waveguide is transversely modified as in the thick half-plane, in the flanged waveguide and in the truncated cone problems. As for the former case, this modified W-H equation can be reduced to a classical vector W-H equation of order two. (see section 1.2.11) 1.1.3. Generalized W-H equations (GWHE) There are three types of generalized W-H equations. The simplest ones are defined by the following equivalent equations:
5
(10a) )())(()()( αααα oFmYFG +−= ++ or, equivalently, by substituting ))(())(( αα mFmY −+ =− (10b) )())(()()( αααα oFmFFG += −+ where )(αm is a known conform transformation that maps the complex plane−α into a new planem − . Eqs. (10) are the W-H equations modeling impenetrable wedge problems or wedge problems with only one propagation constant (Part 3). Handling wedge problems where more propagation constants are present, we have to deal with W-H equations having the following form: (11) ( ) ( ( )) ( ) ( )i i oiA X m X Xα α α α+ −⋅ = + ),...,2,1( ni = where ( )iA α are vectors of order n and where, for the presence of many different propagation constants, many different functions )(αim have to be introduced. A third kind of generalized W-H equations have the form:
(12a) )())(()()(1
αααα oiiij
n
jij FmYXG +−= ++
=∑ ),...,2,1( ni =
Equation (12a) can be rewritten also in the matrix form: (12b) )()(~)()( o α+α=αα −+ FXXG
where:
][X.......
][X][X
)(
n
2
1
α
αα
=α
+
+
+
+X
][F.......
][F][F
)(
3o
2o
1o
o
α
αα
=αF
)](m[Y.......
)](m[Y)](m[Y
)(~
nn
22
11
α−
α−α−
=α
+
+
+
−X ,
)(G...)(G)(G............
)(G...)(G)(G)(G...)(G)(G
)(
nn2n1n
n22121
n11211
ααα
αααααα
=αG
6
1.1.4. Hilbert- Riemann problem The Hilbert-Riemann problem consists in finding an analytic function )(zX whose limiting values )(zX ± from outside of a closed or open curve L satisfy the following equation: (13) LzzBzXzAzX ooooo ∈=− −+ ),()()()( It is easy to see that the Hilbert- Riemann equation eq. (13) is more general than the W-H equation. In fact eq. (13) reduces to eq. (5) when the following conditions are satisfied: - The complex plane z coincides with the plane−α - L is an open curve coinciding with the whole real axis of the plane−α - The function )(αX coincides with )(α+F in the −α upper half plane and with )(α−F in the −α lower half plane. - The function 1)()( −= αα GA is analytic in a strip containing the real axis of the
plane−α 1.1.5. From Wiener-Hopf equations to Fredholm integral equations in the spectral
domain
fig.1.1: Integration paths : 1γ “smile” real axis, 2γ “frown” real axis The Wiener-Hopf equations can be formulated also in terms of Fredholm integral equations [3]. Let us consider eq.5, assuming all the functions to be conventional plus and minus functions, i.e. with no singularities in the respective positive and negative half-planes. By integrating it on the counterclockwise simple closed contour 1γ (“smile” real axis), reported in fig.1, we obtain:
(13) duu
uFduu
uFduu
uFuG o∫∫∫ −=
−−
−+−+
111
)()()()(γγγ ααα
By applying the residue formulas the equation changes into:
7
(14) )(2)()(
01
απαγ ++ =
−∫ jFduu
uFuG
since, due to the regularity properties of the considered minus function, we have
(15) 0)(1
=−∫ − du
uuF
γ α
Similarly we get, choosing the contour 2γ “frown” real axis, reported in fig.1.1 too, as integration path for performing the integral,
(16) 0)(
2
=−∫ + du
uuF
γ α
All these integrals defined on the two considered integration paths 1γ and 2γ , defined on the complex plane, may be rewritten in terms of Cauchy principal value integrals defined on the real axis. For example, starting from eq.16, it yields
(17) 2
( ) ( ). . ( ) 0F u F udu PV du j Fu uγ
π αα α
∞+ +
+−∞= − =
− −∫ ∫
while from the left member of eq.14 we obtain:
(18) 1
( ) ( ) ( ) ( ). . ( ) ( )G u F u G u F udu PV du j G Fu uγ
π α αα α
∞+ +
+−∞= +
− −∫ ∫
By substituting eq.18 in eq.14 and by multiplying by )(1 α−G , it yields
(19) 1 10
( ) ( )( ) . . ( ) 2 ( ) ( )G u F uG PV du j F j G Fu
α π α π α αα
∞− −++ +−∞
+ =−∫
Finally, by combining together eq.19 and eq.17, it yields the Fredholm integral equation valid on the real axis (see also 1.6.3.1, 1.2.11.5):
(20) [ ]
)()()( 1)()(
21)( 1
1
ααα
απ
α +−∞
∞−
+−
+ =−
−+ ∫ oFGdu
uuFuGG
jF
It is important to observe that the equations (20) are Fredholm equations of the second kind, if the quantities 1( )G α± exist and are finite in all the points α of the integration line (including α →∞ ) (see Appendix)
8
As it will be considered in section 1.2.10.3, the previous reduction applies also to the generalized W-H equations (10), (11) and (12). For instance for the equations (12) we
obtain the following not singular integral equation for )(
......)(
)(1
α
αα
+
+
+ =
nF
FF :
(21) ')'()(2
1)(1),(~)(2
1)(1
11 αααπα
ααπ
αγ
dFGj
duuFu
uGGj
F o∫∫ −+
∞
∞−
−+ =
−−⋅+
!
where )(αG is the matrix with entries )(αijG and the entries ),(~ uGij α of ),(~ uG α are:
(22) du
udmmum
uGuG i
iiijij
)()()(
1)(),(~α
α−
=
The reduction of W-H equations to Fredholm equations is very important for many reasons. In the first place the powerful theoretical apparatus developed for Fredholm equations can be apply to the W-H equations too. Moreover in all the applications problems the approximate solutions of the Fredholm equations (e.g. quadrature methods) provide efficient solutions of the W-H equations (1.6.2) if the conditions (21) hold. 1.1.6. Fundamental literature The classic book on the W-H technique is the one by Ben Noble [3] This book, that is based on the Jones approach [2], reports the solutions of the CWHE and the MWHE and some of their fundamental applications. The alternative formulation of the CWHE, called Karp-Weinstein-Clemmow equations, eqs. 6, have been used systematically in the classic book by Weinstein [4]. Compared to the Noble book, the one of Weinstein provides also a complete physical discussion of all of the problems considered. The Riemann-Hilbert problems have been considered in depth in the books by Muskhelishvili [5], Vekua [17] and Gakhov [6]. A more recent book reporting significant progress, not included in the previous classic books, is [7], where, in particular, ch.4 by Kobayashi, ch.s 5 and 6 by Büyükaksoy and Serbest and ch.7 by Lüneburg give a comprehensive presentation of all of the most significant recent literature on the Wiener-Hopf technique. In addition ch.9 by Nosich contains also recent applications of the Riemann-Hilbert problem to the diffraction theory. The theory of the GWHE have been developed by Daniele starting by the generalized Riemann-Hilbert introduced in [17] . The relevant material has been partially published in some papers, among others [9-13]. Furthermore, due to its relevance, the W-H technique is also reported as an important chapter in many books dealing with fields and waves in the presence of different media,
9
and among them we quote the book by Mittra and Lee [8]. This one also reports other analytical techniques, related to the Wiener-Hopf technique, currently used in the literature.
1
8/29/2007 D:\Users\vdaniele\libri\WHnew\chapter 1-2.doc 1.2 Fundamental ideas for solving Wiener-Hopf equations 1.2.1 Introduction Apparently it seems that the presence of the two unknowns )(α+F and )(α−F in eq. (5) of . requires other equations in addition to (1) )()()()( αααα +−+ += oFFFG in order to get the proper solution. However, impressive mathematical literature proved that solving this equation constitutes a closed mathematical problem. This is due to the fact that even though eq.1 has been deduced on the real axis of the α plane, an analytic continuation of it stands and makes it valid everywhere. Dealing with the whole complex plane α , important analytical properties of )(α+F and )(α−F “a priori” known provide the additional information required to solve eq.1. For instance the plus function )(α+F has all its singularities in a lower −α half plane while the minus function )(α−F has all its singularities in an upper −α half plane. This suggests to divide the set S of the involved in eq.1 singularities in two subsets: ∪ −+= SSS where S+ is relevant to the singularities belonging to the upper half plane and S- to the singularities belonging to the lower half plane. Taking into account this fact the fundamental idea is to rewrite eq.1 in such a way that the singularities S+ are present only in the first member and the singularities S- only in the second member. The separation of the singularities will allow us at the same time to obtain the two unknowns )(α+F and )(α−F . An essential role in accomplishing this is played by the kernel )(αG and its inverse 1( )G α− , that in general show singularities in both the half-planes 0]Im[ >α and 0]Im[ <α . The singularities of )(αG and of 1( )G α− are called structural singularities and they always have an important physical meaning. To point out it let us consider some typical W-H geometries indicated in fig.1. In few words, these geometries may be considered as the junction of two (or more) waveguides, or as a single waveguide where geometrical discontinuities are present. The singularities of )(αG , and of its inverse 1)( −αG , are related to the propagation constants of the modes of the involved waveguides. For instance, in the half–plane problem we are dealing with the free space with propagation constant k. This implies that the continuous modes have propagation constants
22 α−± k . In these case the singularities of )(αG and 1)( −αG are the branch points k±=α . Singularities constituted by poles are instead present when we deal with closed
waveguides. The set of the structural singularities define the spectrum of the kernel . This spectrum can be discrete (presence of only poles), continuous (presence of only branch pointes) or mixed (presence of poles and branch pointes). The kernel is rational if the representative matrix has element that are rational functions , meromorphic if the representative matrix
2
has element that are meromorphic functions, quasi rational if the discrete spectrum is finite. In order to avoid the presence of singularities of )(αG , and of its inverse 1)( −αG ,on the real axis α , we always assume the presence of media having at least small losses, on one hand this corresponds to a more realistic case, and on the other it is not restrictive, since the absence of losses may be studied as a limiting case. Remember that with our notations the propagation constant k has a negative (or vanishing) imaginary part and the branch of
22)( ααχ −= k must be chosen such that k=)0(χ . Dealing with a general function )(αf , there are two ways to separate the singularities, corresponding to the concepts of additive and multiplicative decomposition.
1.2.2 Additive Decomposition Given an analytical function )(αf , the problem of the additive decomposition consists in finding two functions )(α+f and )(α−f satisfying the equation: (2) )()()( ααα +− += fff and that are regular in the half-planes 0]Im[ ≥α and 0]Im[ ≤α respectively. To avoid confusion with the Multiplicative Decomposition that will be defined in the following section 1.2.3, occasionally we will use the following symbols to indicate the decomposed functions )(),( αα +− ff of )(αf −− = )}({)( αα ff , ++ = )}({)( αα ff The difficulty of obtaining additive decompositions is the same in the scalar and the matrix case and this problem will be solved by using, generally, the Cauchy integration approach (1.3.1). Alternatively, by taking into account that the plus functions have singularities only in a lower half plane and the minus functions have singularities only in an upper half plane, the Mittag Leffler theorem may turn out to be useful in all of the cases involving the presence of poles only. In the following we will refer to the “additive decomposition” as simply “decomposition”. Example 1
Decompose 22
1)(α
α−
=k
f in plus and minus functions that are regular respectively in
0]Im[ ≥α and 0]Im[ ≤α Taking into account the presence of just the two simple poles k±=α yields:
3
kkf
−=+ α
α ][Res)( , kkf
+−
=− αα ][Res)(
where ][Res k and ][Res k− are the residues in the poles k and –k given respectively by:
k
kfkk
→
−=−=
α
αα21)]()[(lim][Res
, k
kfkk
−→
=+=−
α
αα21)]()[(lim][Res
Of course the problem of the decomposition yields many solutions that differ one with respect to the others in the possible presence of entire functions. We define standard decomposition the one in which the decomposed functions )(α+f and )(α−f either vanish or have the lowest growth order for α going to infinity.
1.2.3 Multiplicative Decomposition or Factorization Given an analytical function )(αf , the problem of the multiplicative decomposition (or factorization) consists in finding two functions )(α+f and )(α−f satisfying the following equation: (3) )()()( ααα +−= fff and such that they, and their inverses, are regular respectively in the half-planes
0]Im[ ≥α and 0]Im[ ≤α . To avoid confusion with the Additive Decomposition defined in the previous section 1.2.2, occasionally we will use the following symbols to indicate the factorized functions
)(),( αα +− ff of )(αf
−− = )]([)( αα ff , ++ = )]([)( αα ff In many cases the Weirstrass factorization of entire functions provides the factorizations involved in many practical situations (see section 1.3). In general we face with many values function or matrices functions and the fundamental idea to factorize a given function ( )f α is based on the logarithmic decomposition. By introducing
[ ( )] ( )( ) Log ff e eα ψ αα = = the decomposition of
( ) [ ( )] ( ) ( )Log fψ α α ψ α ψ α− += = + so yields:
4
(25) ( ) exp[ ( )] exp[ ( ) ( )]exp[ ( )] exp[ ( )]
f α ψ α ψ α ψ αψ α ψ α
− +
− +
= = + ==
This idea involves a commutative algebra and it sure works well for scalar functions. Conversely in the matrix case a not commutative algebra is involved and this idea can be ineffective. The matrix factorization is possible in closed form only in some particular cases that will be considered later. In fact, the general explicit matrix factorization problem is till now an open problem that, in spite of many efforts, up to now does not have a general solution. Example 2 Factorize 22)( αα −= kf in plus and minus functions that are regular respectively in
0]Im[ ≥α and 0]Im[ ≤α The simplicity of the function to be factorized makes it possible to find the explicit solution by inspection: (4) αα −=+ kf )( , αα +=− kf )( Of course the factorization problem yields many solutions that differ for the possible presence of entire functions as long as their inverses are entire as well. We call standard factorization the factorization that involves factorizing functions having algebraic behavior as ∞→α .
1.2.4 Solution of the W-H equation First of all let us consider the case where the unknowns )(α+F and )(α−F are conventional (sect. 1.1) plus and minus functions. The standard factorization of
)()()( ααα +−= GGG yields: (5) )()()()()()( 11 αααααα +
−−−
−−++ += oFGFGFG
Decomposing the second term of the second member in a standard way:
)()()()(1 αααα +−+−− += SSFG o
5
and separating the plus function and the minus one yields: (6) )()()()()()()( 1 ααααααα wSFGSFG =+=− −−
−−+++
Considering now eq. (6) in the whole complex plane α , the analytical function )(αw is regular both in the upper half plane 0]Im[ ≥α (considering the first member) and in the lower half plane 0]Im[ ≤α (considering the second member). Since the two half planes are partially overlapping this means that )(αw is an entire function. The functions )()(),( ααα +−+ oFandFF are generally Laplace transforms. Thus they are vanishing as ∞→α . It means that )(αG has to show a suitable behavior for ∞→α . For instance the factorization )()()( ααα +−= GGG must involve factorized function
( ) and ( )G Gα α− + such that -1( ) ( ) and G Gα α
α α± ± are vanishing as α →∞ .
Looking at the behavior of the standard decomposed and factorized functions, it is possible to ascertain that both the first two members of eq.6 are Laplace transforms. Furthermore, since they are vanishing as ∞→α , it means that the entire function )(αw is vanishing also as ∞→α . Consequently, using the Liouville theorem, )(αw is zero everywhere and this yields the solutions: (7) )()()( 1 ααα +
−++ = SGF , )()()( ααα −−− −= SGF
In order to extend the possibilities of this ingenious technique, it is also possible to deal with generic plus and minus functions in equation (6) that are not vanishing for ∞→α , and consequently the entire function )(αw is not vanishing. In this case the following procedure applies [3, p.37], let us suppose that it could be shown that pSFG αααα <− +++ )()()( as ∞→α , 0]Im[ ≥α qSFG αααα <+ −−
−− )()()(1 as ∞→α , 0]Im[ ≤α
By the application of the extended form of Liouville's theorem it yields that )(αw is a polynomial )(αP of degree less than or equal to the integral part of ),(min qp i.e. )()()()( αααα PSFG =− +++ )()()()(1 αααα PSFG =+− −−
−−
These equations determine )()( αα −+ FandF to within the arbitrary polynomial )(αP , i.e. to within a finite number of arbitrary constants that have to be determined otherwise.
6
1.2.5 Remote Source When the source of the problem is very far (e.g. in the case of plane wave excitation of an open structure or in the case of incident mode excitation of a closed structures) the W-H equations assume the following form: (8) )()()( ααα −+ = FFG where )(α+F and )(α−F are Laplace transforms related to the total fields. The remarkable fact that happens in the presence of remote source is that the W-H equations are homogeneous. It does not produce vanishing solutions of )(α+F and
)(α−F since some Laplace transforms become non-conventional (for example a plus function could have the pole relevant to the source in the upper half-plane 0]Im[ >α ) and the procedure indicated in the previous section cannot be applied any longer. First, to examine the procedures that reduce equation (8) to the conventional W-H equations solved in the previous section, it is important to observe also that the presence of non-conventional Laplace Transforms introduces Bromwich contours (in the inversion equation) that must be suitably located. For instance in the case of a plus function having poles located also in the upper half-plane 0]Im[ >α , the Bromwich contour is not the real axis but an horizontal straight line limiting a lower half-plane containing all of the singularities of the considered plus function. We remember also that plus functions )(α+F (or similarly minus functions )(α−F ) can be introduced as mathematically derived by Laplace transforms of functions having a physical meaning evaluated in α− (true minus and plus functions). For instance:
)()( αα −= −+ XF , )()( αα −= +− XF where )(α+X and )(α−X are true plus and minus functions. We will call induced plus (or induced minus) these functions that are derived by true minus (or plus) functions. To make not homogeneous the homogeneous Wiener-Hopf formulation (8) we consider that a far source introduces a pole oα . It yields a characteristic part in the unknowns
( )F α+ and ( )F α− . In absence of induced plus or minus terms the characteristic parts
present only one term proportional to ( ) 1oα α −− . In general , if also an induced minus
term is present in ( )F α− , the characteristic part of ( )F α− is given by:
( )c
o o
R SF αα α α α− = +− +
and we have:
7
(9) ( ) ( )r
o o
R SF Fα αα α α α− −= + +− +
where ( )rF α− is a conventional Laplace transform that is regular in oα α= ± . Factorizing ( ) ( ) ( )G G Gα α α− += yields:
(10)
1 1
1 1 1 1 1
( ) ( ) ( ) ( )
( ) ( ) ( ( ) ( )) ( ( ) ( )) ( )
o oo o
ro o
o o
R SG F G G
R SG F G G G G w
α α α αα α α α
α α α α α α αα α α α
− −+ + − −
− − − − −− − − − − −
− − − =− +
= + − + − − =− +
Independently of the sign of Im[ ]oα , the first member is regular in the half-plane Im[ ] Im[ ]oα α> and the second member regular in the half-plane Im[ ] Im[ ]oα α≤ so again we have that ( )w α is an entire function. Taking into account the asymptotic behavior of the functions present in (10), Liouville’s theorem yields: ( ) 0w α = and it provides the solutions:
(11) 1 1 1( ) ( ) ( ) ( )o oo o
R SF G G Gα α α αα α α α
− − −+ + − −
= + − − +
1 1( ) ( ) ( ) ( )o oo o
R SF G G Gα α α αα α α α
− −− − − −
= + − − +
To apply eq.(11) we must know the vectors R and S that define the characteristic part
( )cF α− of ( )F α− . We observe that the contribution of the pole oα (or oα− ) in the spectral domain α , represent the asymptotic contribution of the fields as z →−∞ (or z →+∞ for induced minus function). Therefore the vectors R and S can be obtained by physical considerations that provide the knowledge of these asymptotic fields. For instance if the remote source is a plane wave, asymptotic fields are constituted by geometrical optic contributions. Depending on the considered problems, sometimes it turns more simple to consider the
characteristic part ( )c
o o
T UF αα α α α+ = +− +
of the plus function ( )F α+ . Knowing T and
U yields R and S through the W-H equation )()()( ααα −+ = FFG :
( )oG T Rα = , ( )oG U Sα− = Of course the previous equations furnish R and S if T and U are not in the null space of
( )oG α and ( )oG α− . It implies that det[ ( )],det[ ( )] 0o oG Gα α− ≠ .
8
Due to the presence of several reflected rays, sometimes applying the Laplace transform to geometrical optical contribution defined on whole the integration domain 0z−∞ < ≤ (or 0 z≤ ≤ +∞ ) may be difficult. In fact, depending on the considered z-point, the number of the reflected rays may change. To overcome these difficulties it is important to observe that we need only the asymptotic part in the natural domain z of the field components that define the plus and minus functions in the W-H equations. It means that we must consider the geometrical optical contribution only in very far observation points. To avoid any doubt on this procedure we remember that the Laplace transform of the asymptotic components and the Laplace transform of the geometric optic components defined on whole the domain 0z−∞ < ≤ (or 0 z≤ ≤ +∞ ) differs by functions that do not present the poles oα± . It means that these two different Laplace transforms have the same characteristic part. Specific examples considered later will illustrate these considerations adequately. Taking into account the linearity of the W-H equations yields the conclusion that it is not restrictive to reduce W-H equations (.5) to the normal form:
(17) ( ) ( ) ( ) ( )o
RG F X Fα α α αα α+ − −= + =−
where the oα pole is not located on the real axis. According to eqs.(11), the solution of (17) are given by:
1 1( ) ( ) ( )oo
RF G Gα α αα α
− −+ + −=
−
(17) bis
1( ) ( ) ( )oo
RF G Gα α αα α
−− − −=
−
When ( )F α+ presents the pole oα :
( ) ( )o
TF Xα αα α+ += +−
where ( )oG T Rα = , the W-H assumes the form:
(17 ter) ( ) ( ) ( ) ( , )oo
TG X X mα α α α αα α+ −= − ⋅−
where:
( ) ( ) ( )( , ) [ , , ]dG G G tm t If td tα αα αα α
−= ==
−
9
The possible advantage of (17) is the fact that the plus function ( )X α+ is standard and do not present pole oα
1.2.6 Unbounded Plus and Minus Unknowns Case Sometimes the formulation in terms of Wiener-Hopf equations of a specific problem yields the presence of a plus function, that for sake of simplicity we can denote )(ˆ α+F , that even though it is regular in a upper half-plane, and in this sense it is a plus function, it is not a classical Laplace transform, in the sense that it does not show the vanishing asymptotic behavior for ∞→α required for being a proper Laplace transform. For instance )(ˆ α+F may be defined as (18 ) )()()(ˆ ααα ++ = FpF , where )(α+F is a classical Laplace transform and )(αp is a (matrix) polynomial or, more generally, an (matrix) entire function. Following the Wiener-Hopf procedure that yields eq. (6), a non-vanishing entire function
)(αw could be involved. For what concerns the presence of a polynomial factors )(αp , the asymptotic behavior as ∞→α shows that )(αw must be a polynomial. The coefficients of the )(αw polynomial have to be determined by imposing the suitable regularity to )(ˆ)]([)( 1 ααα +
−+ = FpF in order to be a plus function. Some examples can
be seen further on in the followings dealing with some important specific diffraction problems. Similar considerations apply when is the minus functions unbounded: (19) ˆ ( ) ( ) ( )F p Fα α α− −=
1.2.7 Factorized Matrices as Solutions of the Homogeneous Wiener-Hopf Problem It is interesting and important to evaluate the factorized matrices through suitable solutions of the W-H having the same kernel that must be factorized. To this end consider first the homogeneous Wiener Hopf problem: (19) )()()( ααα −+ = XXG where the )(αG is the n-th order kernel matrix. This homogeneous equations yields not vanishing solutions since we are looking for plus and minus functions )(α±X that are regular in 0]Im[ ≥α and 0]Im[ ≤α respectively but are in general not vanishing as
∞→α . This problem is a particular case of the Riemann-Hilbert problem that has been studied by many Russian authors [17]. It is easy to show that n independent solutions
)},(),({ αα −+ ii XX },.....2,1{ ni = of (21) provide the factorized matrices of )( )()( ααα +−= GGG through the equations:
10
(20) 1 2( ) ( ), ( ),........, ( )nG X X Xα α α α− − − −= , 1
21 ),(,),........(),()( −++++ = αααα nXXXG
Of course the existence and uniqueness of n independent solutions )},(),({ αα −+ ii XX of eq. (21) follows from the application of sophisticated theorems to the given matrix )(αG . In all of the cases of practical interest however we can be sure, by physical considerations, that the problem under consideration is mathematically well defined, so in order to factorize )(αG the only problem to face is to get somehow the set
)}(),({ αα −+ ii XX (see also 1.6.3.2). To this end let’s introduce the functions:
(21) ( )( ) ii
p
XF ααα α
++ =
−
where pα is not a structural singularity with Im[ ] 0pα < . For instance notice from (19) that if det[ ( )] 0pG α = ( pα is a pole of 1( )G α− ), it must be
( ) 0i pX α− = and ( ) NullSpace[ ( )]i p pX Gα α+ = otherwise ( )i pX α+ = ∞ . Conversely if 1det[ ( )] 0pG α− = ( pα is a pole of ( )G α ), it must be ( ) 0i pX α+ = and
1( ) NullSpace[ ( )]i p pX Gα α−− = otherwise ( )i pX α− = ∞ .
The homogeneous equation can be rewritten in the form:
(22) ( ) ( ) ( )
( ) ( ) ( ), i=1,2...,ni i p i pi i
p p
X X XG F F
α α αα α α
α α α α− − −
+ −
−⋅ = + =
− −
or:
(23) ( ) ( ) ( ) , i=1,2...,ni
s ii
p
EG F Fα α αα α++⋅ = +−
where ( )i i pE X α−= and the plus ( )iF α+ and the minus ( ) ( )
( ) i i psi
p
X XF
α αα
α α− −
+
−= =
−
are standard Laplace transforms. In the following we will put [ ] [ ] [ ]1 21,0,0,0... , 0,1,0,0... ,... 0,0,0,0...,1t t t
nE E E= = = . Consequently by resolving the W-H equations:
11
(24) ( ) ( ) ( ) ( ), i=1,2...,ni
s ii i
p
EG F F Fα α α αα α++ −⋅ = + =−
provides the determinations of the factorized matrices ( ) and ( )G Gα α− + through the equations (21) ,(20) and (19). It can be shown in the following that changing pα , the factorized matrices differ by a multiplicative constant matrix such that the equation
)( )()( ααα +−= GGG is always satisfied. In fact let’s introduce the factorized matrices , ( )G α− + , 1( , )G α α+ , 2( , )G α α+ where , ( )G α− + are
arbitrary standard factorized matrix of ( )G α and 1( , )G α α+ and 2( , )G α α+ have been determined
with the previous technique by assuming two different values of pα : 1pα α= and 2pα α=
respectively. The Wiener-Hopf solutions of (1.26) yields the following expressions for the vectors 1iX +
and 1iX + that through the equation (1.23) provide the expressions of 1( , )G α α+ and 2( , )G α α+ :
(1.1) 1 1
1,21,2
1,2
( ) ( )i i
G GX R
α αα α
− −+ −
+
⋅=
−
Take into account equations (1.23) and (1.24) to define 2 1( , ) and ( , )G Gα α α α+ + , algebraic manipulations yield: (1.2) 2 1( , ) ( , )G K Gα α α α+ += where the constant matrix K is given by:
(1.3)
11 1 1 12 1 2 2 2 3 2 4
1 1 1 11 1 1 2 1 3 1 4
( ) , ( ) , ( ) , ( )
( ) , ( ) , ( ) , ( )
K G R G R G R G R
G R G R G R G R
α α α α
α α α α
−− − − −− − − −
− − − −− − − −
= ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅
In the following we did not study the problem of the better location of pα in the half-plane Im[ ] 0α < . This problem has relevance in numerical solutions and in these cases we warning that a not suitable choice of pα could have an unsatisfactory numerical expressions for factorized matrices. In fact we introduce with pα an apparent singularity. It does not produces effects on closed form solutions, but in numerical solutions, it increases the possibility to have significant errors in region of the planeα − near pα . To avoid this problem we suggest to define pα in the context of a physical problem related to the factorization of the matrix. For instance in open structures a physical source is constituted by a plane wave having a pole in the planeα − and we can assume
pα in the possible region where this pole is located.
1.2.8 Non-standard Factorizations
12
Many times we are dealing with the factorization of matrices having the form: (25) ( ) ( ) ( ) ( )G R M Tα α α α= ⋅ ⋅ where ( ) and ( )R Tα α are rational matrices and the factorized matrices ( )M α− , ( )M α+ of the matrix ( ) ( ) ( )M M Mα α α− += are explicitly known. We can write formally the factorization: (26) )(~ )(~)( ααα +−= GGG where: (27) ( ) ( ) ( )G R Mα α α− −= ⋅" , ( ) ( ) ( )G M Tα α α+ += ⋅"
)(~ α−G , )(~ α+G and their inverses present algebraic behavior at infinity without being true minus and plus functions because of the presence of a finite number of simple poles located in the respective regularity half planes. This type of factorization has been referred to in literature as weak factorization [7], [14]. It is possible to eliminate the offending poles by using many different techniques. (see 1.2.6). In essence the ideas are two. First we rewrite the W-H equations
( ) ( ) ( ) , Im[ ] 0s oo
o
RG X Xα α α αα α+ −= + <−
in the form:
1 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( )so oo o
o o
R RG X G G X G Gα α α α α α αα α α α
− − − −+ + + + − + + − = + − − −" " " " "
For the presence of offending poles present in ( )G α+" and ( )G α−
" , the first member and the second member are not respectively a plus and a minus function. We can reduce these terms to plus and minus function by subtracting the characteristic ( )C α−
" and
( )C α+" due to the offensive poles present respectively in Im[ ] 0α > and in Im[ ] 0α < :
1
1 1 1
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
oo
o
s oo
o
RG X G C C
RG X G G C C w
α α α α αα α
α α α α α α αα α
−+ + + + −
− − −+ − + + + −
− − − =−
= + − − − = −
" " " "
" " " " "
Now the first member is plus since ( )C α−" eliminates the offending poles of ( )G α+
"
located in Im[ ] 0α > . At the same time the second member is minus since ( )C α+"
eliminates the offending poles of ( )G α−" located in Im[ ] 0α < . Consequently the function
13
( )w α is entire and for the algebraic behavior of the two members it vanishing. It follows:
1 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( )oo
o
RX G G G C G Cα α α α α α αα α
− − − −+ + + + + + −= + +
−" " " " " "
1 1( ) ( ) ( )
( ) ( ) ( ) ( ) ( )o os
o
G G G RX G C G C
α α αα α α α α
α α
− −+ + +
− + + + −
− = − + +−
" " "" " " "
The coefficients of the characteristics in the poles can be obtained by forcing that ( )X α+ and ( )sX α− are plus and minus functions (see 1.5.1.1) . In practical situations it can
happen that 1( )G α±+" and/or 1( )G α±
−" can have a set of equal offensive poles (see for
example (1.6.1.6)). Even though we have 1, ,( ) ( )G Gα α−
+ − + −⋅ = 1" " , in these case sometimes this fact can introduce trouble for the presence of not vanishing null spaces; then it could be useful to resort the other idea that introduces a suitable entire function. To illustrate this idea for the factorization (25), it can be observed that is not restrictive to put
( ) 1R α = and to face with the factorization of (28) ( ) ( ) ( )G M Tα α α= ⋅ . In fact if we had to factorize ( ) ( ) ( )G R Mα α α= ⋅ , we could use the procedure that will indicate by reversing the role of the function plus and the function minus.
Putting ( )( )( )
PTdααα
= where )(αP and )(αd are, respectively, matrix and scalar
polynomials in α , eq.(19) becomes:
(29) 1( )( ) ( ) ( ) ( )( )
PM X M Xdαα α α αα
−+ + − −=
The vector ˆ( ) ( ) ( ) ( )M P X Xα α α α+ + += is a plus function. By following the same considerations as in section (1.2.4), the factorization of )(αd yields:
(30) 1ˆ ( ) ( ) ( ) ( ) ( )
( )X d M X wd
α α α α αα
−+− − −
+
= = ,
where )(αw is an unknown polynomial vector that can be determined by considering the first and the third member of (30):
(31)
1 1 1 1
1
ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) a
X P M X d P M wd P M w
α α α α α α α αα α α αα
− − − −+ + + + +
−++
= = =
=∆
14
where
(32) )()(
)(1
αα
α∆
=− aPP
has been introduced, with )(α∆ and )(αaP the determinant and the adjoint matrix of
)(αP respectively. Let us suppose that all the zeroes of )(α∆ are simple, and let us indicate with i−α (i=1,2..) the zeroes located in the upper half-plane 0)Im( >α . . Of course ( ) 0id α+ − ≠ Since ( )X α+ is a plus vector, it follows from eq.31 that : (33) 1( ) ( ) ( ) 0a i i iP M wα α α−
− + − − = (i=1,2..) It follows that 1( ) ( )i iM wα α−
+ − − must satisfy the equation: (34) ( ) ( )i i i iw c M uα α− + −= (i=1,2..) where iu is in the null space of )( iaP −α and ic is an arbitrary constant. Eq.29 allows the evaluation of the matrix polynomial coefficients in terms of the constant ic , (i=1,2..). By changing the values of these constants, if the factorization of the rational matrix ( )G α is well posed, it is possible to see that there are n and only n independent
)(αw : )}(),......(),({ 21 ααα nwww satisfying eq.34 so that we have obtained n independent ( )X α+ and ( )X α− given by:
(35) 1( )( ) ( ) ( ) ( )( ) a
dX P M wαα α α αα
−++ +=
∆
(36) ( )( ) ( )( )
wX Mdαα αα− −
−
=
according to the results of section 1.2.7, These function provide the factorization of
( ) ( ) ( )G G Gα α α− += .
Example : Factorize the rational matrix 2 2
2 21( )
1
AB
jqG
jq
ααα
++= where q, A and B are
real. We have a rational matrix with ( ) 1M α =
15
It is possible to obtain:
2222
2222
)()(
)(αααα
α++++
=BBjq
AjqBP
2222
2222
)()(
)(αααα
α++−+−+
=BBjq
AjqBPa
22)( Bd +=αα , ))(()( 22222222 αααα qqABB ++++=Λ
jB=−1α , 2
222
21 q
qABj+
+=−α
Eq.(30) implies that the degree of )(αw is 1, so it is possible to rewrite )(αw in the following form:
dcba
w++
=αα
α )(
The null spaces of )( 1−αaP and )( 2−αaP are respectively the vectors:01
and 1/ qj−
. By
eq.(28), this means that: 01
)( 11 Cw =−α , 1/
)( 22
qjCw
−=−α where 1C and 2C are
arbitrary constants. This yields to the following four equations in the a, b, c, d unknowns.
11 Cba =+−α , 01 =+− dcα , qjCba 22 −=+−α , 22 Cdc =+−α
( )2
222
2
1
1
q
qABj
qjC
jB
Ca
+
++−
+−= ,
( )2
222
2
1
11
q
qABj
qjC
jB
CjBCb
+
++−
++= ,
2222
22
1
1
qABqBqjC
c+−+
+= ,
2222
22
1
1
qABqBqBC
d+−+
+=
By substituting }0,1{ 21 == CC or }1,0{ 21 == CC it is possible to obtain two independent solutions of eq.(28):
0
1)(
2
222
2
222
111 q
qABj
q
qABj jBB
BW
+
+
+
+ +−−
+−+
=
α
α
16
2222
2
2222
2
112
1
1
1
1)(
2
222
2
222
qABqBqj
qABqBqB
BqBq
jB
W q
qABq
q
qABq
+−+
++
+−+
+
+−−
+−= +
+
+
+
α
α
α
By substituting in eq.(29) it is possible to obtain the factorized matrices:
( )( )( )
2 2 2
2
2 2 2
2
2 2 2 2
2 2 2 2
1
1
1 1
1( )
0
q B A q
q
B A q
q
j B A q j q
B q B A q B j BqG
j
B
α
αα
+
+−
+
+
+ + +
− + + + + −=
−− +
( )( )( )
( )( )( )
2 2 2
22 2 2
2
2 2 2 2 2 2 2 2
1
1
1 10
( )q B A q
qq B A q
q
B q B A q B A q j q
q B jG
Bq jBBq
q B j
α
αα
α
α
++
++
+
− + + + + − +
−=
− +− −
−
1( )G α−
− and 1( )G α−+ here not reported, are respectively minus and plus functions too.
Another non-standard factorization of the kernel )(αG so yields: (20) )(ˆ )(ˆ)( ααα +−= GGG where )(ˆ α−G and )(ˆ α+G are (together with their inverses) conventional plus and minus functions showing non-algebraic behavior for ∞→α . In this cases the W-H procedure leads to an entire function )(αw , non-vanishing for ∞→α . The deduction of this function is generally very difficult to obtain [16]., so that, generally speaking, it appears preferable to avoid the presence of the entire function )(αw and to obtain standard factorized matrices )(α−G and )(α+G working on the non-standard matrices )(ˆ α−G and
)(ˆ α+G
17
If ( )G α is a scalar , factorized functions ˆ ( )G α± having exponential behavior qe α± as α →∞ arises from factorization equations that make use the Weierstrass factorization of entire functions (1.3.3.2). In these case the offending behavior are immediately eliminate by putting (see (1.3.3.2)) :
ˆ( ) ( ) qG G e αα α± ±= ∓ Conversely when the kernel are matrices this problem is considerable more difficult to solve. When the given matrix ( )G α commute with a polynomial matrix ( )P α , we can
obtain factorized matrices ˆ ( )G α± by using the method indicated 1.5.10.2. Unfortunately
with the exception of some (but impotant) cases, ˆ ( )G α± present offending behavior as α →∞ . Again the key idea to remove the unacceptable behavior at α →∞ is to rewrite the factorization equation in the form: (137) )(ˆ)()()(ˆ)(ˆ)(ˆ)( 1 ααααααα +
−−+− == GRRGGGG
where )(αR is an arbitrary rational matrix commuting with ( )P α ( and also with ( )G α ) .
For example, it is possible to choose: (138) 2
1 2( ) ( ...)1 ( )oR x x x Pα α α α= + + + + where all the coefficients ix (i=0,1,…) are arbitrary constants. It is possible to factorize
)(αR by the same procedure that factorize ( )G α (sect.1.5.10.2). It yields: )(ˆ)(ˆ)( ααα +−= RRR and also )(ˆ)(ˆ αα +− RandR have non-algebraic behavior as ∞→α . As we have at disposal arbitrary constant ix in )(αR , we can choose them such that )(ˆ)(ˆ αα −− RG
and )(ˆ)(ˆ αα ++ RG have algebraic behavior. If it can be accomplished, taking into account the commutativity of the involving matrices we rewrite: (140) )(ˆ)(ˆ)()(ˆ)(ˆ)(ˆ)(ˆ)( 1 αααααααα ++
−−−+− == GRRRGGGG
This reduces the factorization of )(αG to that of the rational matrix
)()()(1 ααα +−− = ii RRR . This factorization can be obtained with method described in
1.5.5 and it yields the final equations:
18
(141a) )()(ˆ)(ˆ)( αααα −−−− = iRRGG (141b) )(ˆ)(ˆ)()(ˆ αααα ++++ = GRRG i The factorization thus obtained could be weak, since the factorized matrices could have offending poles, but it is not difficult to overcome this problem (sect.1.5.9). Conversely, the big difficulties of this procedure are related to the search for the unknowns ix
(i=0,1,…). In fact, even though the factorized matrices )(ˆ α−R and )(ˆ α+R can be evaluated in closed form, the equations that must be satisfied by the unknowns ix (i=0,1,…). are not linear. Some detailed examples are reported in [16],[52],[53]. A general method is described in ….. 1.2.9 The matching of the singularities in cases involving a discrete spectrum
Let’s consider a rational matrix ( ( ), ( ), ( ), ( )A B dα α α δ α polynomials) ( )( )( )
AGdααα
=
having with its inverse 1 ( )( )( )
BG ααδ α
− = poles that are all simple. Let’s suppose that in
the W-H equation (17) Im( ) 0oα < :
( ) ( ) ( ) ( )o
RG F X Fα α α αα α+ − −= + =−
( )G α ( ( )d α ) presents only the (minus) poles (zeroes) located in the upper half plane
Im( ) 0nα > in 1, 2,..m mα α= = with residue ( )G mR α . We obtain :
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0G m m G m m
m mm o m
R F R FRG F X wα α α αα α α αα α α α α α
+ ++ −
⋅ ⋅− − = − + = =
− − −∑ ∑
It provides the plus and the minus solutions ( )F α+ and ( )X α− in terms of the unknowns ( ) ( 1,2...)mF mα+ = :
1 ( ) ( )( ) ( ) G m m
m m o
R F RF G α αα αα α α α
− ++
⋅= + − −
∑
( ) ( )( ) G m m
m m
R FX α ααα α
+−
⋅=
−∑
19
In order to obtain the unknowns ( ) ( 1,2...)mF mα+ = , we must observe that the minus poles (zeroes) ( 1,2,..)n nβ = of 1( )G α− ( ( )δ α ) located in Im( ) 0nβ > are offending for the plus functions ( )F α+ . Different situations arise. For instance if det[ ( )]G α is constant and the involved matrices are of second order the poles ( 1, 2,..)n nβ = coincides with
the poles ( 1, 2,..)n nα = . Since 1 1det[ ( )] costantdet[ ( )]
GG
αα
− = = we eliminate the
offending poles by imposing:
1 ( ) ( )( ) 0, 1, 2,G m mn
m n m n o
R F RG nα ααα α α α
− + ⋅+ = = − −
∑ ; it implies the equations:
( ) ( ) 0, 1,2,G m m
m n m n o
R F R nα αα α α α
+⋅+ = =
− −∑
that allow to evaluate ( ) ( 1,2...)mF mα+ = In general we have:
( ) ( )( ) ( ) G m m
m m o
R F RG F α αα αα α α α
++
⋅⋅ = +
− −∑
Taking into account that:
( )det[ ( )]det[ ( )]
N
GB
δ ααα
=
where N is the order of the matrix, it follows that usually det[ ( )] 0mG β = . So the vector ( ) ( )G m m
m n m n o
R F Rα αβ α β α
+⋅+
− −∑ is not vanishing but it belongs to the null space of the
matrix ( )nB β . For instance if the null space of ( )nB β is the vector n
uβ , we have:
( ) ( ) , 1, 2,..n n
G m m
m n m n o
R F R C u nβ βα αβ α β α
+⋅+ = =
− −∑
It introduces additional unknowns n
Cβ but usually in a well posed problem the number of the offending poles nβ provides a number of equations that is sufficient to determine both the unknowns ( )mF α+ and the constants
nCβ .
The presence of additional unknowns is regrettable. To avoid it , we will present in the section 1.5.4.2.1 a slight different method The matching of the singularities can be extended to meromorphic matrices (1.5.7.1)
20
1.2.10 Mathematical Requirements of the Existence and Uniqueness Theorem In the previous section it has been pointed out how the use of homogeneous solutions to obtain the solution of the Wiener-Hopf equations is an useful tool. From a mathematical point of view, the existence of homogeneous solutions is related to the study of the existence and uniqueness of the solution of the Wiener-Hopf equations. This is a topic that has been extensively studied in the past; concerning the vector cases, e.g. the fundamental work by Gohberg and Krein [18] is particularly important. These aspects of the theory of the Wiener-Hopf equations are very important, even though difficult to deal with. However, they do not suggest constructive factorization techniques for general matrices. 1.2.11 Solution of the Equations Associated to the Wiener-Hopf Equations. In this section we briefly report the ideas or the relevant literature allowing the solution of the equations associated to the Wiener-Hopf equation considered in section . Many times these procedures aim to reduce the equations under study to the well studied Fredholm or Volterra integral equations. The equation 3 of is the basic W-H equation written in the natural domain. It may be reduced to a repeated Volterra equation with the procedure described in [3, p.230]. However, this procedure involves the factorization kernel and it appears to us not too appealing for practical applications. The Karp-Weinstein-Clemmow equations 6 of have been used systematically in the classic book by Weinstein [4]. Their solution is based on the factorization idea and the basic procedure is also reported in [3, p.220]. The modified W-H equations have been introduced and solved by Jones [19], [20]. Their solutions, as well as many important examples of their applications, are reported in the book by Noble [3, chap.6] and, using different methods from those of Jones, in the book by Mittra and Lee [8]. Besides the ‘ad hoc’ procedures already presented in literature it is worth observing that the (transversely) modified W-H equations: (23) )()()()()()( αααααα oFFFHFG =+−+ −++ may always be reduced to a CWHE equation by using the following procedure: Changing α with α− in (23) yields: (24) )()()()()()( αααααα −=−+−+−− −++ oFFFHFG
21
Equations (23) and (24) together constitute a classical W-H equation having as plus and minus functions the unknowns:
−
=−
++ )(
)()(
αα
αFF
F ,
−−=
−
+− )(
)()(
αα
αF
FF
These equations have the following matrix form: (25) )()()()( αααα oFFFG += −+ where:
(26a)
−+−
−−
=)()()()()(
1)()(
1)(ααααα
αα
αHHHGG
HG
G
(26b)
−+−−
=)()()()(
)()(
1)(αααα
αα
αoo
oo FHFG
FG
F
The (longitudinal) modified W-H equations (L>0): (27) )()()()exp[)()( αααααα oFFFLjFG =++ −+ may be reduced to a CWHE too. In order to accomplish such reduction, first of all we observe that the entire function )(αF can be considered as a plus function, since it is regular in the half-plane 0]Im[ ≥α and moreover has algebraic behavior as +∞→]Im[α . Conversely, the entire function )(αα Fe Lj− can be considered as a minus function since it is regular in the half-plane 0]Im[ ≤α and has an algebraic behavior as −∞→]Im[α . Considering the eq. (27) and the equation obtained multiplying (27) by ( )Ljα−exp the second order Wiener-Hopf system yields: (28) )()()()( αααα oFFFG += −+ where:
(29)
=
− 0)(
)( Lj
Lj
eeG
Gα
ααα ,
=
++ )(
)()(
αα
αFF
F ,
−
−= −−
− )()(
)(α
αα α Fe
FF Lj
22
( )
( )0
oo
FF
αα
=
For what concerns the Hilbert-Riemann eq. (12) of , there is vast literature describing the relevant theory. For instance, the book by Noble [3] reports in section 4.2 of chapter IV the fundamental ideas allowing the explicit solution of these equations. Hilbert problem is more general than the Wiener-Hopf problem for many different reasons. Noble discusses them in detail arguing even that there are some pieces of evidence for preferring the Wiener-Hopf technique over the Hilbert-Riemann technique when both of these techniques apply. Wiener-Hopf equation equivalent integral equations in the spectral domain have already been discussed in section .5. For instance, it has been shown that these equations can always be reduced to the classical integral equation of Fredholm type. A major discussion in this section concerns the generalized Wiener-Hopf equations (.3): (30) )())(()()( αααα oFmFFG += −+ These equations were obtained by Daniele studying wedges diffraction (see part 3)1. Before developing a technique for solving these equations it is important to ascertain whether they, as it happens for the classical W-H equations, constitute a closed mathematical problem, and to this purpose we can refer to chapter 4 of [17]. Even though this reference deals only with generalized Hilbert problems and it introduces transformations )(αm mapping closed contours onto themselves, it yields many relevant to the existence and uniqueness of the solution results useful in developing effectively the theory of equations of .3. However, it must be recognized that the role of the factorization is neglected in [17], and, as it often happens in purely mathematical approaches, no constructive methods are reported there to explicitly solve these problems. Recall that, in the eq. (30), )(α+F is a conventional plus function in the α -plane (i.e. regular in half plane 0]Im[ ≥α ), ))(( αmF− is a minus function in the m-plane (i.e. regular in half plane 0]Im[ ≥m ) and )(α+oF is a known function. If αα =)(m eq.(30 ) reduces to a classical W-H equation that can be solved with the decomposition and the factorization techniques. From a purely conceptual point of view, the decomposition (additive or multiplicative) is the key to obtain the solution for the generalized Wiener-Hopf equations too. However, whereas the classic additive decomposition is easy to obtain by the reported in the 1.3 chapter procedure, the generalized decomposition is very difficult and in general requires the solution of an integral equation. On the contrary, concerning the factorization, there is no conceptual difference with the classical case because we can use the same kind of mathematical manipulations that will be introduced later.
1 A more careful analysis of the literature has shown that these equations have been introduced, but not studied, in the ex. 5.15, page 219 of Noble’s book too.
23
1.2.11.1 Generalized decomposition The problem of the generalized additive decomposition consists in finding two functions
)(α+F and ))(( αmF− satisfying the equation: (31) )())(()( ααα +− += FmFF where )(αF is an analytical function and )(αm is a suitably chosen mapping, such that they are regular in the 0]Im[ ≥α and 0]Im[ ≤m half-planes, respectively. As it happens for the classical decomposition, the level of difficulty is the same in the scalar and the vector cases but the generalized decomposition cannot be obtained by simple formulas. However, when )(αF is a conventional Fourier transform, it constitutes a closed mathematical problem.
fig. 2.1: Integration paths In fact, with reference to the left part of fig. 2.1, we have:
(32) )(''
)'(0''
)'(1
απαααα
αααα
γ +
∞
∞−
++ −−
==− ∫∫ FjdFdF
where the second member integral must be considered a Cauchy principal value integral. Let us call )(mα the inverse function of )(αm , and suppose that it is regular at least in a strip of the complex plane m containing the real axis. In this case it is possible to rewrite eq. (31) in the following form: (33) ))(()())(( mFmFmF αα +− += Considering the integration paths reported in the central part of fig.2.1, we have
(34) ∫∫∫∞
∞− +++ +
−=
−=
−))(('
'))'('(
''
))'('('
'))'('(
22
mFjdmmmmFdm
mmmFdm
mmmF απ
αααγγ
where the third member integral must be considered a Cauchy principal value integral.
24
Introducing the mapping )(αm in eq. (34) it yields:
(35) )()'')()'(
)'('')()'(
)'('33
απαααα
ααααα
αγγ +
+ +−
=− ∫∫ Fjd
ddm
mmFd
ddm
mmF
where 3γ and '3γ are respectively the mapping of the line 2γ and the real axis of the m-plane onto the α -plane (fig.2.1 left) and the second member integral must be considered a Cauchy principal value integral. Let us now suppose that: a)- 3γ has the same direction as 1γ b)- there are no singularities of )(αF between 3γ and 1γ . In these conditions we can deform the line 3γ into the real axis of the −α plane and eq. (35) reduces to:
(36) )('')()'(
)'('')()'(
)'( απαααα
αα
αααα
γ +
∞
∞−
+ +−
=− ∫∫ Fjd
ddm
mmFd
ddm
mmF
where the second member integral must be considered a Cauchy principal value integral and the path γ has been reported in the right part of fig.2.1. Subtracting (32) from (36), it yields: (37)
∫∫ −=
−
−−
+∞
∞− ++ γα
αααα
παα
αααααπα '
')()'()'(
21')'(
'1
')()'(1
21)( d
ddm
mmF
jdF
ddm
mmjF
Equation (37) is a Fredholm integral equation of the second kind. Its solution solves the generalized decomposition problem. In particular, when αα =)(m , the second term in the first member does vanish and (37) reduces to the classical Cauchy decomposition formula (1.3.1) :
(38) ''
)'(2
1)( ∫ −=+ γ
αααα
πα dF
jF
We can now suppose that one or both the conditions a) and/or b) are not satisfied. Even in these cases the problem remains again mathematically closed, but it becomes more difficult to solve. For instance if 3γ does not have the same direction as 1γ , both terms of the first member integral have the same sign, and the integral equation (37) becomes singular.
25
Let us observe that if 0)( =αF , we deal with homogeneous equations, and the only solution vanishing at infinity of this equation is the trivial one 0)( =+ αF . It means that in this case we are looking for the generalized decomposed functions of 0 that are vanishing as ∞→α : (39) )())((0 αα +− += FmF directly implies that: (40) 0)( =+ αF , 0))(( =− αmF 1.2.11.2 Generalized Factorization The problem of the multiplicative decomposition consists in finding two functions )(α+f and ))(( αmf− satisfying the following equation: (41) )())(()( ααα +−= fmff where )(αf is an analytical function and )(αm a known mapping, such that they, and their inverses, are regular in the 0]Im[ ≥α and 0]Im[ ≤m half-planes respectively. As it happens for the classical factorization, this problem is much more difficult in the vector case compared to the scalar case. However, many times we can perform the generalized factorization by using the same expedient used for the classical factorization. 1.2.11.3 Solution of Generalized Wiener Hopf Equation (30) The only one wave number case If we are able to obtain the factorization of )())(()( ααα +−= GmGG as indicated in eq.41, we can rewrite eq.30 in the form: (42) )())(())(())(()()( 11 αααααα oFmGmFmGFG −
−−−−++ +=
Applying to )())((1 αα oFmG −
− the generalized decomposition procedure: (43) )())(()())((1 αααα +−
−− += SmSFmG o
it yields: (44) ))(())(())(()()()(0 1 αααααα mSmFmGSFG −−
−−+++ −−−=
The previous equation can be considered as one of the possible generalized decompositions of the 0 function. From eq. (40) the solutions so follow:
26
(45) )()()( 1 ααα +
−++ = SGF , ))(())(())(( ααα mSmGmF −−− −=
In conclusion, it must be observed that also the Generalized Wiener Hopf equation reduces to a Fredholm integral equation. In fact, by using the same procedure leading to eq. 37, we can show that the eq. (30) is equivalent to the integral equation: (46)
∫∫ −=
−
−−
+−∞
∞− +
−
+1
'')()'(
)'()(2
1')'('1
')()'()'()(
21)(
11
γα
ααααα
παα
ααααααα
πα d
ddm
mmFG
jdF
ddm
mmGG
jF
Even though this integral equation could be singular, we can always make use of the powerful solving techniques indicated for example in [5]. The general case Considering a general case, the problem may be modeled in terms of equations (11b) of .3, that involve more mappings )(αim . This case too has been considered in chapter 4 of [17], in the Hilbert problem framework. Even for these equations the concept of additive generalized decomposition can be introduced. For instance the generalized factorization of the matrix kernel )(αG is expressed by: (47) )()()( ααα +−= GGG
$$
where the following factorized matrices may be obtained by finding n independent solutions
)(....
)()(
)( 2
1
α
αα
α
+
+
+
+ =
in
i
i
i
F
FF
F and
))((....
))(())((
)( 22
11
α
αα
α
nin
i
i
i
mF
mFmF
F
−
−
−
− =$
(i=1,2,..n)
of the homogeneous equation: (48) )()()( ααα −+ −= ii FFG
$
In particular we have: ),(,),........(),()( 21 αααα −−−− = nFFFG
$$$$,
(49) 1
21 ),(,),........(),()( −++++ = αααα nFFFG
$
In order to obtain the solution of the system
27
(50) )())(()()(1
αααα oiiij
n
jij FmYFG +−= ++
=∑ (i=1,2..n)
it is possible to extend to this case too the theory developed for the generalized Hilbert problem in chapter 4 of [17], reducing the generalized W-H equations to the following
not singular integral equation for )(
......)(
)(1
α
αα
+
+
+ =
nF
FF :
(51) ')'()(2
1)(1),(~)(2
1)(1
11 αααπα
ααπ
αγ
dFGj
duuFu
uGGj
F o∫∫ −+
∞
∞−
−+ =
−−⋅+
$
where )(αG is the matrix with entries )(αijG and the entries ),(~ uGij α of ),(~ uG α are:
(52) du
udmmum
uGuG i
iiijij
)()()(
1)(),(~α
α−
=
and
(53)
)'(')()'(
1.........
)'(')()'(
1
)'(')()'(
1
)'( 22
22
11
11
αααα
αααα
αααα
α
onn
nn
o
o
o
Fddm
mm
Fddm
mm
Fddm
mm
F
−
−
−
=$
Reducing the GWHE to Fredholm integral equations is important for two main reasons: first it shows that the solution of GWHE poses a well-defined mathematical problem, then it allows well known numerical schemes of solutions to solve the problem. 1.2.11.4 Reduction to a Classical W-H Equations Problem. A more important way for solving GWHE is constituted by the introduction of a suitable mapping )(ααα = that reduces these equations to classical W-H equations, e.g. this is possible dealing with wedge problems involving only one wave number (see Part 3) while, dealing with the presence of many different mappings )(αim , in general, it seems not possible. However, the introduction of the mappings )( ii ααα = , reducing the functions )(α+if and ))(( αimy −+ to functions )( iif α+ and ))(()( αα iii myf −= +− , regular in the two complementary half-planes 0]Im[ >iα and 0]Im[ <iα respectively, can be very useful to simplify the solution of these equations. For example we can solve in closed form
28
equations that involve quasi-rational kernels; these kernels are meromorphic kernels presenting a finite number of poles. 1.2.11.5 The Solution of Wiener-Hopf Equations in the Framework of the
Integral Equation Theory In section .5 it has been ascertained that the Classical WH equations
)()()()( αααα oFFFG += −+ may reduce to the solution of the following integral equation
(54) [ ]
∫∫ −=
−
−+ −∞
∞−
+−
+1
)()(21)( 1)()(
21)( 1
1
γ αα
παα
πα du
uuFG
jdu
uuFuGG
jF o
This formulation allows an approximate general procedure of solution that will be discussed in section 1.6.2. The factorization of the matrix kernel )( )()( ααα +−= GGG , provides immediately the solution of the integral equation (54). In fact we observe that the procedure of section .5 applied to the equation:
)()()()()()( 11 αααααα oFGFGFG −−−
−−++ +=
yields directly the solution:
(55) ∫ −=
−−
++1
)()(2
1)()(1
γ απαα du
uuFuG
jFG o
since the unknown integral
[ ]∫∞
∞−
++−+
−− du
uuFuGG
j αα
π)(1)()(
21 1
is vanishing.
The solution (55) is equivalent to the solution obtained in 1.2.4. When explicit factorizations of )(αG are not possible, if the integral equation is of Fredholm type, it can be solved with classical schemes. If )(uG has a suitable behavior as ∞→u , the presence of a kernel that is not singular when α=u assures this circumstance, since the
kernel [ ]
αα−
−+−+
uuGG 1)()(1
is a compact operator. In general, the required behavior of
)(uG as ∞→u can be obtained by a preliminary factorization of a suitable scalar involved in )(uG .
29
The formulation of W-H equation as integral equations points out also an important class of matrix kernel )(αG yielding closed form solutions. These are the rational matrices or more generally the meromorphic matrices involving, with their inverses, a finite number of poles (quasi rational matrices). Let us suppose the presence of only simple poles in )(αG , denoting with jβ those located in the 0]Im[ >α half plane. The residue theorem yields:
(56) [ ]
∑∫ −=
−
− +−
∞
∞−
+−
j j
jj FRGdu
uuFuGG
j αββα
αα
π)()()(1)()(
21 11
where jR is the residue of )(αG in jβ . Consequently eq. (54) can be rewritten in the form:
(57) )()()(
)()( 11 αααββ
αα +−+−
+ =−
+ ∑ oj j
jj FGFR
GF
Supposing the presence of only simple poles iα in )(1 α−G , located in the half plane
0]Im[ >α , it requires that the vectors )()(
ioj ij
jj FFR
ααββ
++ −−∑ must be in the null space
of the matrices )()( 1ii G ααα −− . As it will be discussed later, this allows to obtain the
unknowns )( jF β+ that provide, through eq. (57) a full representation of )(α+F . Dealing with generalized W-H equations involving many )(αjm functions a question arises: do rational matrices or quasi rational matrices )(αG yield closed form solutions? The method used in this section for classical W-H equations do not apply for GWHE. However it will be showed in the next sections that the problem of solving GWHE with quasi rational kernel always reduces to a problem of generalized scalar decomposition of functions )(αf : (58) )())(()( ααα +− += fmff i This fact remarks the importance of the introduction of rational (or quasi rational) approximation of the matrix kernel )(αG in order to obtain efficient solution of arbitrary W-H equations. 1.2.11.6 Solution of Vector W-H Equations with Quasi Rational Matrix
Kernels In this section we will show another technique for solving W-H equations where the matrix kernels are quasi rational. This procedure is interesting since it applies also to generalized Wiener Hopf equations that cannot be reduced to Classical W-H equations.
30
However, in this case, the problem is reduced to generalized decomposition of scalar functions. In order to consider the more general case we start from the generalized equation: (59) )()]([)()( αααα +−+ +=⋅ oiiii FmFFg (i=1,2..n) where the vector )(αig is defined by the i-row of the quasi rational matrix kernel )(αG . In the following we denote with riβ and sα the poles of )(αig and )(1 α−G located in the half plane 0]Im[ >α . The last equation can be rewritten in the form:
(60) )]([)(
)()(
)()( αβαβ
αβαβ
αα iir r
riir
roi
ri
riiri mF
FrF
FrFg −
++
++ +
−
⋅−=−
−
⋅−⋅ ∑∑ (i=1,2..n)
where irr is the residue of )(αig in irβ The generalized scalar decomposition:
(61) )()]([1 ααβα +− +=
− riiriir
fmf (i=1,2..n), (r=1,2..)
yields (i=1,2..n), (r=1,2..) (62)
0)]([)()]([
)()()](1[)()(
=+⋅−=
=−⋅−−
−⋅
−+−
++++
∑
∑αβα
αβαβα
αα
iiirrir
iri
roiirriri
iri
mFFrmf
FFrfFg
where the vanishing of all the members follows from the generalized decomposition of the 0 function expressed as sum of plus and minus generalized functions
0)]([)()]([)()()](1[)()( =−⋅+−⋅−−
−⋅ −+−++++ ∑∑ αβααβαβα
αα iiririr
irir
oiriririri
i mFFrmfFFrfFg
It follows the representation of the unknown )(α+F in the form:
(63)
+⋅−−
−= ∑ +++−
+ ])()()](1[)()( 1
roiririri
ri
FFrfGF αβαβα
αα
31
Taking into account that )(α+F is a plus function, it follows that the vector
+⋅−−∑ +++ ])()()](1[
roiririri
ri
FFrf αβαβα
belongs to the null space of )()( 1ss G ααα −−
(s=1,2,..). This allows to evaluate the unknowns )( rF β+ , (r=1,2..), to complete the representation of )(α+F . The minus function )]([ αii mF − is given by: (64) )()]([)]([ riri
ririii FrmfmF βαα +−− ⋅= ∑ i=1,2,…n
and it does not introduce additional equations to evaluate the unknowns )( irF β+ . To conclude we also observe that the remarks of 1.2.11.4 can reduce the generalized decomposition (61) to a classical decomposition one. 1.2.11.7 The Method of Moments for Solving General W-H Equations An approximate, but very general, method for solving the W-H equations is the moment method. In this section we briefly show the use of this method for solving the generalized equations 11 of sect. .3. In order to do that let us represent the unknowns )(α+jX in the following form: (65) )()( αα j
rr
jrj CX ++ Ψ= ∑
where the set of the “expansion” functions )(αjr+Ψ (r=1,2,..) is known and suitably
chosen. Substituting in the eq..11a yields:
(66) )()]([)()(1
αααα iiij
r
n
j r
jrij FmYCG +−=Ψ ++
=∑ ∑ (i=1,2,..n)
We can obtain the eliminations of the minus variables )]([ αii mY −+ by considering the previous equation in the im -plane:
(67) )]([][)]([)]([1
iiiiij
r
n
j r
jriij mFmYmCmG ααα +−=Ψ ++
=∑ ∑ (i=1,2,..n)
By using the Parseval theorem, the projection of the “test” functions )( i
js m+Φ yields:
(68) 0)()( =−⋅−Φ +
∞
∞− +∫ iiiijs dmmYm
Consequently, for every value of i and s the following equation hold:
32
(69) si
n
j r
jr
ijsr NCM =∑∑
=1
where the unknown coefficients may be so computed in terms of the projection integral defined by the choice of the expansion basis functions:
(70) iij
riijijs
ijsr dmmmGmM )]([()]([()( αα +
∞
∞− + Ψ⋅−Φ= ∫
(71) iiiijssi dmmFmN )]([()( α∫
∞
∞− + ⋅−Φ=
If N is the total number of the expansion or test functions, the linear system (69) is a system of Nn× linear equations having the Nn× unknowns: j
rC . The success of the moment method depends on the suitable choice of the set of the expansion and test function. This choice must be accomplished on the basis of physical considerations in order to use very few expansion functions, the smallest set of them to achieve the required accuracy.
1.2.12 Important mappings for dealing with W-H equations 1.2.12.1 The 22 ατχ −= o mapping In many propagation problems the functions involved in the Wiener-Hopf equations depend on the complex variable α , and, typically, even on the 22 ατχ −= o function. This happens when in the considered physical problems the geometrical discontinuities are present in isotropic media with wave number oτ . It is not restrictive to suppose that
oτ is the free space propagation constant k, and so in the following we occasionally assume ko =τ .
The function 22 ατχ −= o presents the two branch points oτα ±= and, after having considered two arbitrary branch lines, we define the principal branch that the gives
oτχ = as 0=α . This is a compulsory choice, due to the physical existence of the Green function. In fact, if we chose two branch lines not crossing the real axis, if follows that for the principal branch 0]Im[ <χ on all the real axis. The solution of the Wiener-Hopf equations must be proper, and this means that, in their analytical expressions, it must be assumed the principal branch of 22 ατχ −= o . To understand the properties of the
mapping 22 ατχ −= o it is convenient to trace on the plane−α the lines where
0]Im[]Im[ 22 =−= ατχ o and 0]Re[]Re[ 22 =−= ατχ o .
33
fig.2.2: Mapping 22 ατχ −= o
With reference to fig.2.2, these lines consist in arcs of equilateral hyperbolas. If the branch lines are the vertical lines as indicated in fig.2.2, the plane−α is divided in sub regions A,B,C,D that allow to individuate the principal branch for every point considered in this plane. For instance starting from 0=α we can reach one point of the region B using different paths, in particular, if we choose a path that crosses the arc where
0]Im[]Im[ 22 =−= ατχ o we for reasons of continuity must assume 0]Re[ >χ , 0]Im[ >χ . Choosing another path that links the point 0=α to a point of the region B
yields the same conclusion. In fact if we assume the path that crosses the arc where 0]Re[]Re[ 22 =−= ατχ o we first reach the region D where 0]Re[ <χ , 0]Im[ <χ .
Successively we can reach one point of the region B by crossing the assumed vertical branch line where there is a discontinuity of the principal branch. It produces again
0]Re[ >χ , 0]Im[ >χ . One must observe that the use of other branch lines could modify the sign of 0]Im[ <χ for the principal branch. It is remarkable that when the branch lines are coincident with the arcs hyperbolas where 0]Im[ =χ , we obtain a principal branch having 0]Im[ ≤χ everywhere. In this circumstance the principal branch is called proper branch and the relevant cut plane is called proper sheet. In the following we always assume these branch lines(see fig. 2.3)
34
fig.2.3: Standard Branch lines in the plane−α . In order to point out effectively the branch lines, in fig.2.3 oτ has been assumed j−1 . It is important to observe that on the lower lip and on the upper lip of the branch line −b ,
the proper branch of 22 ατχ −= o is real and respectively positive and negative. Similarly on the upper lip and on the lower lip of the branch line +b , the proper branch
of 22 ατχ −= o is real and respectively positive and negative. By reminding that the result of ]arg[z is always between π− and π+ , this figure
illustrate also the values of
+
o
jτ
χαarg on the end points of the lips of the branch lines.
These values will be considered in the next section. 1.2.12.2 The wo cosτα −= mapping In the cases where the mapping 22 ατχ −= o is present, it turns out useful to introduce the angular complex variable w defined through the mapping : (72) )cos( awo += τα There are many reasons for introducing this mapping. For instance some specialized analytical method introduce representations different from the Laplace transform. In particular the Malyuzhinets method introduces the Sommerfeld representations that may be considered as spectral representations in the w-plane.
35
Furthermore, the introduction of the mapping (72) may help to handle and theoretically manipulate functions that may be conversely hard and obscure if defined only in the
plane−α . The parameter a is arbitrary but, for sake of convenience, as it will be shown later on, in the following it will be assumed always π . A detailed study of the mapping (73) wo cosτα −= is very important. In particular it is important to ascertain the properties of the function
22 ατχ −= o in the w-plane. For instance, since the plus function )(α+F has only the
branch line corresponding to the branch point oτα = , if we cut the branch line corresponding to the other branch point oτα −= , even though the variable χ changes of sign with the discontinuity, )(α+F must remain continuous. Given α , the inverse of mapping of (73) introduces infinite values of w given by:
(74) πτ
χατ
χαπτ
χαα )12(arglog)log()( −+
++
+−=−
+−= njjjjjw
ooo
,
,....2,1,0 ±±=n where the argument of a complex number z is assumed:
ππ +≤<− ]arg[z
No additional branch points with respect those due to 22 ατχ −= o arise from the
logarithm. In fact, o
jτ
χα + is never vanishing as α is a finite point.
( 2220 αταχα +=→=+ oj ) So, in the following, we assume as branch lines of the mapping the standard branch lines constituted by the arcs hyperbolas where 0]Im[ =χ (fig. 2.3). In the following we assume as principal value )(αw the following branch:
(75) ])log(,)log(,2
[arg)( πτ
χαπτ
χαπτ
χαα ++
−−+
−−>+
=ooo
jjjjjIfw
where the proper branch of 22 ατχ −= o and the principal value of ]log[z have been assumed:
0]Im[ ≤χ , ]arg[log]log[ zjzz += , ( ππ +≤<− ]arg[z )
36
and the function ],,[ cbaIf gives b if a evaluates to true, and c if it evaluates to false. The presence of the function ],,[ cbaIf has been introduced in order to avoid that the
discontinuity of o
jτ
χα +arg across the branch point oτα −= (see fig. 2.3) produces a
discontinuity on )(αw . When α is real we have that our principal branch is related to the principal part of the arccos:
(76) )arccos()arccos()(oo
wταπ
ταα −−=−= 0]Im[ =α
Furthermore, since: (77) wo sinτχ ±= on the image rw of the real axis 0]Im[ =α , we have:
jwo
oo
ejwjwj
wjwjwjw
jw
∓∓ log)sinlog(cos
))sin(coslog())sin(cos
log()(
−=−=
=−±+−−=−±+−
−= ππτ
ττα
This means that only choosing wo sinτχ −= yields the identity ww =)(α . Finally, it is important to observe that for the principal value: (78) παα −−=− )()( ww Equation (75) yields the representation of fig.2.4 of the proper sheet in the w-plane
37
fig.2.4: Image of the proper plane−α on the w-plane
In this figure the images wr and wb± respectively of the real axis and of the standard branch lines of the plane−α (fig. 2.3) are reported. With the mapping defined by eq.73, both the proper and improper sheets of the
plane−α lie down on the w-plane. For instance, we can observe that the segment of the w-real axis defined by: 0≤≤− wπ is the image of the segment ),( oo ττ− belonging to the proper sheet, whereas the segment of the real axis defined by ππ −≤≤− w2 is the image of the segment ),( oo ττ− belonging to the improper sheet. Since all the w-plane is image of the proper, of the improper and of other infinite sheets arising from the inversion of the mapping (73), we must be very careful when we will rewrite in the plane−α equations that have been obtained in the w-plane. Conversely, we cannot assume as correct for all the points of the w-plane the properties that are valid only for the images of the proper sheet. For instance, the function χ in the w-plane is expressed by: ww ooo sin)cos1( 2222 ττατ −=−=− At the same time, this one looks an even function of w (second member) and an odd function of w (third member). The reason of this paradox is due to the fact that the second member is valid only for values of w that are images of proper sheet. On the contrary, by changing w in –w, we are forced to go into the improper sheet and according to the analytical continuation of woo sin22 τατ −=− , )cos1( 22 wo −τ changes of sign. In the following (1.4 and Part 3) it turns out very useful to deal with functions of w defined in all the w-plane. In general we can always do it through analytical continuation, but, we must however remember that, as it happens in the previous paradox, the analytical continuation may involve improper values of χ .
38
The complete mapping (73) is depicted in fig.2.5. The wave number "' ooo jτττ += has been assumed complex, with imaginary negative part ( )0"<oτ . The quadrants 1,2,3 and 4 of the proper sheet in the α plane are denoted by the same number encapsulated in a box, S. For instance, the point ooo ϕτα cos−= ( πϕ << o0 ) has as proper image
oow ϕ−= .
fig.2.5: The complete mapping wo cosτα −=
An Application: Asymptotic evaluation of the integrals: 2 2
( ) j k yj xcI f e e dααα α
∞ − −−
−∞= ∫
The integral 2 2
( ) j k yj xcI f e e dααα α
∞ − −−
−∞= ∫ is very important since it arises in diffraction
problems in stratified structures. Here x and y stands as Cartesian coordinates of an observation point. In cylindrical coordinates we have:
(78) cossin
xy
ρ ϕρ ϕ
==
Putting o kτ = and using the mapping indicated in 1.2.12.2:
(1.1) cossin
k wk w
αχ= −= −
I
II
arcTan0
0
ττ
−=∆
39
yields the integral:
(1.2) cos( )ˆ ( )w
jk wc sr
I k f w e dwρ ϕ−= ∫
where : (1.3) ˆ ( ) ( cos )sinsf w f k w w= − and wr is the image of the real axis in the w plane− (fig. 2.4). Let’ s consider the points sw where the function: (1.4) ( ) cos( )q w j w ϕ= − has vanishing derivative: (1.5) '( ) sin( ) 0s sq w j w ϕ= − − = These points are called saddle points since they are saddle points of the real functions
(1.6) 1 2 1 2
1 2 1 2
( ) ( , ) Re[ ( )] sin( )sinh
( ) ( , ) Im[ ( )] cos( )cosh
u w u w w q w w w
v w v w w q w w w
π ϕ
π ϕ
= = = + −
= = = + −
where 1w and 2w are real and imaginary part of w (F-M).: (1.7) 1 2w w jw= + There is only one saddle point relevant located in the images of the proper planeη − ( 0swπ− ≤ ≤ ) (fig.2.4). It is defined by:
1 2, 0s s sw w wπ ϕ π ϕ= − + ⇒ = − + = Let’s consider in the w plane− the Steepest Descent Path (SDP) crossing the saddle point (F-M). This line is defined such that :
(1.8) ( ) ( )sv w v w= , ( ) ( )su w u w≤ .
40
Taking into account that ( ) cos( ) ( ) 0, ( ) 1s s s sq w j w j u w v wϕ= − = − ⇒ = = − bears to the following definition of the SDP: (1.9) 1 2( , ) 1v w w = − 1 2( , ) 0u w w ≤ or: (1.10) 1 2cos( ) cosh( ) 1w wπ ϕ+ − = , 1 2sin( )sin 0w wπ ϕ+ − ≥ Equations (3) yields: (1.11) 1 2( )w gd wπ ϕ= − + + where ( )gd t is the Gudermann function defined by:
(1.12) 1( ) cos[ ]sgn( )cosh
gd t arc tt
=
We oriented the SDP according to the increasing of the values of 2w
Looking at the fig. 2.4 , notice that if 2πϕ > points of the SDP go in images of the
improper sheet of the planeη − We have:
(1.13) cos( ) ( )ˆ ˆ( ) ( )jk w jk k u wSDP s sSDP SDP
I k f w e dw ke f w e dwρ ϕ ρ ρ− −= =∫ ∫
where on the points of the SDP ( )u w is negative or vanishing in the saddle point.: (1.14) ( ) 0u w ≤ on the SDP In the far field evaluation the parameter is 1kρ >> so that, taking into account that
( ) 0u w ≤ and that there is present a rapidly decreasing exponential ( )k u we ρ , we observe
41
that the major contribution to the integral SDPI is due to points very near the saddle point
sw . If there are not poles of ˆ ( )f w near the saddle point, it yields the approximate evaluation:
(1.15) cos( ) ( )ˆ ˆ( ) ( )jk w jk k u wSDP s s sSDP SDP
I k f w e dw ke f w e dwρ ϕ ρ ρ− −= ≈∫ ∫
To evaluate the integral we introduce the mapping: (1.16) 2( ) ( )sq w q w s− = − Of course when w moves on the SDP s is real. It will be assumed positive when 2 0w > and negative when 2 0w < . It yields:
(1.17)2 2( ) '( ) '(0) '(0)k s k sk u w
SDPe dw e w s ds w e ds w
kρ ρρ π
ρ∞ ∞− −
−∞ −∞= ≈ =∫ ∫ ∫
From (8)
(1.18) '( ) 2dwq w sds
= − or 2'( )
dw sds q w
−=
Applying the Hospital’s rule it yields: 2'(0)
"( ) '(0)s
wq w w
−= or taking into account that "( )sq w j=
42'(0) 2 2"( )
j
s
w j eq w
π−= = = ±
Looking at fig we observe that the sign of the argument dwds
is positive so that
(1.19) 4'(0) 2j
w eπ
= It yields the following approximation of SDPI defined in (7)
(1.20) ( / 4) ( / 4)2 2ˆ ( ) ( cos )sinj k j kSDP sI k e f k e f k
k kρ π ρ ππ ππ ϕ ϕ ϕ
ρ ρ− − − −≈ − + = −
42
Eq. (1.22) cannot be used when poles of ˆ ( ) ( cos )sinsf w f k w w= − are near the saddle point and the approximation (9) does not hold longer. In this case we must use a more general technique indicated in (F-M) or (V-S) If the characteristic of ( )f α in the pole coso okα ϕ= − near
cos( ) coss k kα π ϕ ϕ= − − + = is:
o
Rα α−
We obtain:
2
( / 4)1 (2 cos )2 2( cos )sin
2cos2
o
j kSDP
o
F kI e k f k R
kρ π
ϕ ϕρπ ϕ ϕϕ ϕρ
− −
+−
≈ − + +
where ( )F z is the Kouyomjian-Pathak transition function defined by:
)]4
([)]4
(exp[22)(2 πππ −−−= ∫
∞ − zArguzjzjdseezjzFz
jszj
where the principal branch of z is: 4
5]arg[4
3 ππ<<− z
The previous equations provide the evaluation of the integral on the SDP. We can relate this evaluation to the original one on the path wr by warping the contour path wr in (1.6) into the SDP and take into account the eventual singularity contributions located in the region between the SDP and wr . Sometimes is preferable to study the warping directly in the planeα − . In this plane the SDP do not change if we change ϕ with ϕ− . The fig. illustrates the location of the SDP in the planeα − .
43
The versus of the SDP in the planeη − is in according to the versus of th SDP in the w-plane
z
1
D:\Users\vdaniele\libri\WHnew\chapter1-3.doc 1.3 Functions decomposition and factorization
1.3.1. Decomposition The general formula for decomposing a complex function vanishing as α →∞ as a sum of a minus and a plus function )()()( ααα +− += FFF is given by the Cauchy equation [3, p.13]:
(1) 1
1 ( ') 1 1 ( ')( ) ' ( ) '2 ' 2 2 '
F FF d F dj jγ
α αα α α απ α α π α α
∞
+ −∞= = +
− −∫ ∫ , 0]Im[ ≥α
where the rightmost integral must be considered a Cauchy principal value integral.
fig.3.1: Integration paths
Alternatively, in the second member integral in eq. (1) the hook integral symbol
may be used, where the hook on the integral sign indicates that the chosen integration path, for example 1γ reported in fig.3.1, is the deformed real axis that passes below the pole αα =' . Occasionally, in the following we will call the line 1γ the �smile� real axis. Similarly, the minus decomposed term can be obtained:
(2) 2
1 ( ') 1 1 ( ')( ) ' ( ) '2 ' 2 2 '
F FF d F dj jγ
α αα α α απ α α π α α
∞
− −∞= − = −
− −∫ ∫ 0]Im[ ≤α
z
2
where the rightmost integral must be considered a Cauchy principal value integral. Occasionally, in the following we will call the line 2γ the �frown� real axis. Alternatively, as in the previous case, we may use the following symbol:
where the minus sign accounts for the clockwise sense of the integration path. Equations (1) and (2) hold in the respective regularity half-planes. By analytical continuation of the integrals evaluation it is possible to obtain )(α+F and )(α−F everywhere, and, if the integrals are evaluated in closed form, they directly provide the analytical expressions of
)(α+F and )(α−F everywhere. On the other hand, if we use a numerical integration for the Cauchy integral in eq.1 the jump at the discontinuity from 0]Im[ ≥α to 0]Im[ ≤α can be taken into account by subtracting the residue in the pole α , i.e. )(αF− (fig.3.2). It yields the following representation of )(α+F as 0]Im[ ≤α :
(3) 1 1 ( ')( ) ( ) '2 2 '
FF F dj
αα α απ α α
∞
+ −∞= − +
−∫ , 0]Im[ ≤α
fig.3.2: Modification of the integration path for evaluating )(α+F as 0]Im[ ≤α
The same result can be achieved by applying equation (2), valid for 0]Im[ ≤α , and the equation:
)()()( ααα −+ −= FFF
z
3
A remarkable simplification arises when we are dealing with even )(αF , i.e. )()( αα −= FF . In fact, by changing 'α with - 'α in the integral in eq.2 it yields:
(3) )()( αα −= +− FF Several proofs of the fundamental decomposition formulas can be found in [3]. In the following we report a simple one that is based on the Cauchy formula. Let�s consider the upper region limited by contour 1γ+ ∞+Γ = Γ ∪ where ∞+Γ is the half-circumference
located at ∞ . Since the plus decomposed ( )F α+ is regular in this domain and vanishing on ∞+Γ , the Cauchy formula provides the following equation:
1
1 1
1 1
( ') ( ')1 1( ) ' '2 ' 2 '
( ') ( ')1 1' '2 ' 2 '
( ') ( ') ( ')1 1 1' ' '2 ' 2 ' 2 '
F FF d dj j
F Fd dj j
F F Fd d dj j j
γ
γ γ
γ γ
α αα α απ α α π α α
α αα απ α α π α α
α α αα α απ α α π α α π α α
+
−
+ ++ Γ
−
−
Γ
= = =− −
= − =− −
= − =− − −
∫ ∫
∫ ∫
∫ ∫ ∫
0]Im[ ≥α
where being ∞−Γ is the half-circumference located at ∞ in the lower half-plane,it is
1γ− ∞−Γ = Γ ∪
The integral ( ')1 '
2 'F d
jα α
π α α−
−
Γ−
−∫ is vanishing since being 0]Im[ ≥α the integrand is regular in
the interior of the region limited by −Γ Example 1: Let us decompose the function 22)( ατα −== kF ( ))0( kF = where k represents a propagation constant. Since the function is not vanishing for α going to infinity, eq. (1) does not directly
apply. However, by rewriting 22
22 1)()(α
αα−
−=k
kF we can decompose through
eq.1 the function:
)()(122
ααα
+− +=−
SSk
,
where:
z
4
(4) ∫ −−=+
1
112
1)(22γ απ
α duuukj
S
In order to find a closed form expression for the plus decomposed function )(α+S , we first evaluate the rightmost integral assuming real values of α , and, afterwards, we extend the resulting expression to the whole complex α -plane by analytic continuation. Before evaluating the integral in eq. (4), it is important to study in detail the two-valued function 22 α−k , having the two branch points k±=α (fig. 3.3) (see 1.2.12.1).
fig.3.3: Standard branch lines of the 22 α−k function
On the arcs of the equilateral hyperbole both the two opposite branches have either vanishing imaginary or vanishing real part according to the indications of fig.3.3. In order to keep the branch 22 ατ −= k ( k=τ when 0=α ) single valued we introduce artificial barriers called branch lines. When we cross the branch lines, the single-valued function τ changes with discontinuity assuming two opposite values on the two lips of the branch line. The branch lines originate in the branch point where the two branches of
the two-valued function 22 α−k assume the same value 022 =−±= k
kα
α . Branch
lines are arbitrary, with the only limitation not to cross the real axis, and the standard branch lines are the arcs of the equilateral hyperbole where 0]Im[ =τ (fig.3.3). With this choice it may be ascertained that 0]Im[ ≤τ in the entire −α complex plane (1.2.12.1). Another way to achieve the purpose of the branch lines described above is to imagine that the −α plane consists of two sheets superimposed on each other: by cutting the sheets along the standard branch lines and by imagining that the lower edge of the bottom sheet is joined to the upper edge of the top sheet, it yields that the proper sheet is defined by the sheet where the function 22 α−k assumes the value τ , while the improper sheet is the
other sheet where 22 α−k assumes the value -τ .
z
5
fig.3.4: Deformation of the integration path
To evaluate the integral (3):
∫ −−=+
1
112
1)(22γ απ
α duuukj
S
we deform the integration path 1γ into the line ∪ −+= λλλ made of the two lips of the branch line Γ relevant to the positive branch point k+ (fig.3.4). On the lip +λ the proper branch is positive and real valued, whereas on the lip −λ the proper branch is
negative and real valued. Taking into account that ∫∫−+
−=λλ
this yields:
∫∫
∫
−+
=−−−
+−−
=
=−−
=+
λλ
γ
απαπ
απα
duuukj
duuukj
duuukj
S
112
1112
1
112
1)(
2222
221
∫− −−
−=λ απ
duuukj
11122
By deforming the half contour −λ into the half- straight line 1λ indicated in fig.3.4 ( 1λ belongs to the straight line that includes the points 0 and k) it yields:
∫∫ −−−=
−−−=
−+
1
111111)(2222 λλ απαπ
α duuukj
duuukj
S
By putting xku = on 1λ we evaluate the integral with MATHEMATICA:
z
6
(5) ∫ ∫ =−−
−==−−
−=∞
+1 1 222
1
1
11111)(λ απαπ
α dxkxxj
duuukj
S
2222
)cos()cos(
απ
α
απ
απ
−
−=
−
−=
kk
Arc
kk
Arc
The closed form thus obtained shows that )(α+S is a conventional plus function, since its Taylor series expansion about the point k−=α leads to:
(6) ])[(15
)(23
1)cos()( 3
3
2
222kO
kk
kk
kkk
ArcS ++
++
++=
−
−=+ α
πα
πα
παπ
α
α
and the dominant term near the branch point k=α is given by
(7) kkkk
kArc
S ≈−
≈−
−=+ α
ααπ
α
α ,2
1)cos()(
22
Due to the relevance of the applications of this plus function, we report here some of its alternative expressions:
(8) k
jjkj
kjjk
kArc
S αττπατ
αττπαπ
α
α −=
+++−
=−
−=+ log1
)()(log1)cos(
)(22
(9) k
jjkj
kjjk
kArc
SS αττπατ
αττπαπ
α
αα +=
−+−−
=−
=−= +− log1)()(log1)cos(
)()(22
In all of these expressions 22 ατ −= k is the value of the proper branch ( 0]Im[ ≤τ ) and the functions cosArc and log have to be considered as principal values. Sometimes it is interesting to express the decomposed )(, α−+S in the angular complex plane w defined by the mapping: (10) wk cos−=α wk sin−=τ This mapping (10) is particularly important for studying angular regions, as it has already been seen in the previous section 1.2.12.2 and it will be considered again in section 1.4. In the plane w the decomposed )(, α−+S assume the form (see (76) of 1.2):
z
7
(11a) wk
wwkSk
kArc
Ssin
)cos()cos(
)(22 παπ
α
α =−=−
−= ++
(11b) wk
wwkSk
kArc
kk
ArcS
sin)cos(
)cos()cos()(
2222 ππ
απ
απ
απ
α
α −−=−=
−
−−=
−= −−
Eq.s (11) may be obtained in a straightforward way by using an ingenious suggestion made by Ament [3, p.47]. From (8) and (9) the decomposition of 22 ατ −= k follows: (12) +− +=−= ττατ 22k with:
(13a)k
jjkj
kjj
kArck ατ
πτ
ατατ
πτ
π
ααατ −
=+++−
=−−
=+ log)()(log
)cos()(
22
(13b)k
jjkj
kjj
kArck ατ
πτ
ατατ
πτ
π
ααατ +
=−+−−
=−
=− log)()(log
)cos()(
22
It is often very important to study the asymptotic behavior of the decomposed function as ∞→α . By taking into account that
±∞→±= ]Re[, αατ asj
it follows that, independently of the sign of ]Re[α ,
(14) ]2log[])2log[()( 1
kkα
παα
παατ −=−±≈ ±
± as ∞→α
Decomposition of an even function The decomposed ( ) and ( )F Fα α+ − can be rewritten:
1 1 ( )( ) ( ) . .2 2
F uF F PV duj u
α απ α
∞
+ −∞= +
−∫
1 1 ( )( ) ( ) . .2 2
F uF F PV duj u
α απ α
∞
− −∞= + −
−∫
z
8
Since ( ) = ( )F Fα α− , changing u with -u yields:
1 1 ( ) 1 1 ( )( ) ( ) . . ( ) . . ( )2 2 ( ) 2 2 ( )
F u F uF F PV du F PV du Fj u j u
α α α απ α π α
∞ ∞
− +−∞ −∞
−= + + = − + = −
− − − −∫ ∫ Numerical Decomposition The numerical quadrature of the decomposition formulas requires some care since it can introduce spurious singularities in the decomposed functions. For instance if h and A represent respectively the discretization step and the truncating parameter on the quadrature of the integral we get the approximation:
/
/
1 1 ( ) 1 ( )( ) ( ) . . ( )2 2 2 2
A h
i A h
F u h F hiF F PV du Fj u j h i
α α απ α π α
∞
+ −∞=−
= + ≈ +− −∑∫ hi α≠
We observe that in the exact formula the singularities of ( ) F α located in upper half plane are exactly compensated by the analytical continuation of the exact values of the integral. It does not happens when the integral is approximate by the sum. Moreover with the discretization process + ( ) F α presents offending singularities located in
, / ,.. /hi i A h A hα = = − + . The first circumstance is physiological and in some sense acceptable since by increasing the accuracy of the numerical quadrature yields to compensate the singularities of ( ) F α located in upper half plane. Conversely the second fact yieds a regrettable result and to overcome this problem we rewrite the decomposition formula in the form:
1 1 ( ) 1( ) ( ) . . ( )2 2 2
1 ( ) ( ) 1 1lim . . ( ) lim . .2 2
M M
M MM M
F uF F PV du Fj u
F u FPV du F PV duj u j u
α α απ α
α απ α π α
∞
+ −∞
− −→∞ →∞
= + = +−
−+ −
− −
∫
∫ ∫
The last integral is vanishing. Thus we get:
1 1 ( ) ( )( ) ( ) lim2 2
M
MM
F u FF F duj u
αα απ α+ −→∞
−= +
−∫
Consequently the quadrature of the integral yields:
/
/
( ) ( )1( ) ( )2 2
A h
i A h
F h i FhF Fj h i
αα απ α+
=−
−≈ +
−∑
By this way the offending poles , / ,.. /hi i A h A hα = = − + disappear. Taking into account that Cauchy formula provides the exact representation:
z
9
( )1 1( ) ( ) . .
2 2F uF F PV du
j uα α
π α∞
++ + −∞
= +−∫
we can increase the accuracy of the evaluation of a plus function ( ) ( )oF Fα α+ +≈ by using the equation:
/
/
( ) ( )1( ) ( )2 2
A ho o
oi A h
F h i FhF Fj h i
αα απ α
+ ++ +
=−
−≈ +
−∑
We can obtain an alternative quadrature formula by assuming ( ) ( ) ( )F F Fα α α−= " " where
( )F α−" is an arbitrary minus factor of ( )F α vanishing as α →∞ .
Now we can write:
( )1 1 1 1
( ) ( ) ( )( ) ( ) ( )( )2 ( ) ( )F u F F uF u F u F uF uj F du du du F du
u u u uγ γ γ γ
απ α α
α α α α−− −
+
−= = = +
− − − −∫ ∫ ∫ ∫" " "" " ""
Since ( )F α−
" is minus and vanishing as α →∞ , the last integral is zero . Beside
( ) ( )1
( ) ( ) ( ) ( ) ( ) ( )F u F F u F u F F udu du
u uγ
α α
α α∞− −
−∞
− −=
− −∫ ∫" " " " " "
and we get:
( )( ) ( ) ( )1( )2
F u F F uF du
j uα
απ α
∞ −+ −∞
−=
−∫" " "
It yields the quadrature equation:
( )/
/
( ) ( ) ( )( )
2
A h
i A h
F h i F F h ihFj h i
αα
π α−
+=−
−≈
−∑" " "
It is important to observe that if ( )F α−
" is assumed such that ( ) ( )F Fα α+=" " is a plus function, this approximation does not produces spurious poles Example 1bis Let� consider the numerical decomposition of the function considered in example 1:
)()(122
ααα
+− +=−
SSk
we have:
z
10
( )/
/
( ) ( ) ( )( )
2
A h
ai A h
F h i F F h ihSj h i
αα
π α−
+=−
−≈
−∑" " "
where we have assumed:
1( ) ( )F Fk
α αα+= =
−" " , 1( )F
kα
α− =+
"
In the figures � are reported the absolute errors ( ) ( )( )( )
aS SeSα αα
α+ +
+
−= on the real
axis and on the half-lines ( 0) and ( 0)j x x k j x xα α= ≥ = + ≥ located on the regular half plane of the plus functions (decomposizione_tau2.nb)
z
11
1 j xα = +
31 10 , 500, 0.01k j A h−= − = =
Case of meromorphic functions It must be observed that in many cases we are dealing with meromorphic functions. In these cases instead of using the general formula (1) the Mittag Leffler decomposition, which is based on the following Theorem [21,p 175], turns out to be very useful. Theorem 1 Let us suppose that the meromorphic function )(αF has simple poles
,....,......,, 21 kααα ( )0( ≠kα with residues ],....[Res],......,[Res],[Res 21 kααα . If , except
in the poles, ∞→→ ααα asF ,0)(lim , the following expansion holds:
(15) ∑∞
=
+
−+=
0
11][Res)0()(k kk
kFFααα
αα
If in addition the meromorphic function )(αF is proper, e.g. it satisfies:
∞→→ αα asF ,0)(lim , eq.(15) simplifies and becomes:
z
12
(16) ∑∞
= −=
1
1][Res)(k k
kFαα
αα
Eq. (16) allows the decomposition of a meromorphic proper )(αF by separating the poles k+α having positive imaginary part, 0]Im[ >+kα , from the poles k−α having negative imaginary part, 0]Im[ <−kα . It yields: (17)
)()(1][Res1][Res1][Res)(111
αααα
ααα
ααα
αα +−
∞
= ++
∞
= −−
∞
=
+=−
+−
=−
= ∑∑∑ SSFk k
kk k
kk k
k
with:
(18) ∑∑∞
= +++
∞
= −−− −
=−
=11
1][Res)(1][Res)(k k
kk k
k SSαα
αααα
αα
In many applications it is necessary to know the asymptotic behavior of the decomposed functions as ∞→α . For any finite number of poles, we obtain: (19) 1
, )( −−+ ≈ ααS
The case of an infinite number of poles is more critical. In order to achieve this, let us suppose that :
(20) )0(,)]1(1[][Res <+= pk
Oka pkα and )]1(1[
kOkbk +=α as ∞→k
It follows that:
(21) ∑∑∞
=
∞
= +++ −
+−
−−
=11
1}11][Res{)(k
pp
k kk kb
kakb
kaSαααα
αα
In the previous expression, it may be shown that the dominant term ∞→αas is the last
sum ∑∞
= −1
1k
p
kbka
α. The inverse Laplace transform of this term yields ∑
∞
=
−−1k
zkbpekaj .
Now we have [8, p.11]1:
(22) )1(
1
)()1( +−∞
=
−∑ +Γ−≈− p
k
xkbp xbpajekaj as +→ 0x .
By applying the Laplace transform, we return to the α-plane and eqs. (21,22) yield: 1 Mittra Lee limited his proof to the case 01 <<− p
z
13
(23) pp
jbpjaS −
+−+ −
+Γ−≈)(
1)1()( )1(
αα
The decomposition formula has been applied many times. So in the literature some slight modifications to both proofs and applications can be found. These modifications are discussed in many texts and in particular in Noble [3]. To conclude this section we remark again that the general decomposition formula (1) is valid both in the scalar and in the matrix case. Decomposition by using rational approximants of the function In the case of rational functions, the Mittag-Leffler expansion involves a finite number of terms (partial fraction expansion). Consequently we can accomplish the decomposition of arbitrary functions ( )f α by using partial fraction expansion of rational approximants of ( )f α . A very popular technique to obtain rational approximants is based on the Pade representation (see 1.6.1.2). However in many worked examples the Pade approximants do not have a good accuracy (decomposizione_tau3.nb) so it is better to rationalize the function with an interpolation approximant method described in 1.6.1.3 . For instance by using this technique, for the function
)()(122
ααα
+− +=−
SSk
we obtained (decomposizione_tau4.nb) the error:
( ) ( )( )( )
aS SeSα αα
α+ +
+
−=
illustrated in the following figures.
z
14
z
15
1.3.2. Factorization 1.3.2.1. General formula for the scalar case The fundamental idea to factorize a given function )(αG is based on the logarithmic decomposition. By introducing (24) )()]([)( αψαα eeG GLog == the decomposition of )]([)( ααψ GLog= so yields: (25) )()()()]([ αψαψαψα +− +==GLog where from (1) :
(26) ''
)]'([2
1)(1∫ −
=+ γα
ααα
παψ dGLog
j )()()( αψαψαψ +− −=
By taking into account that in the scalar case:
z
16
(27) )()()()()( αψαψαψαψαψ +−+− == + eeee one obtains the factorization formulas: (28) )()()( ααα +−= GGG where:
(29) ]'
)]'([2
1exp[)(1∫ −
=+ γα
ααα
πα dGLog
jG , )()()( 1 ααα −
+− = GGG
If )(αG is even, we obtain the simplification )()( αα −= +− GG The third member of eq. (27) is valid even in the matrix case, if the matrices defined by eq. (26) )()( αψαψ +− and commute, but, unfortunately, )()( αψαψ +− and do commute only for very few applications. The integral defining )(α+G in (29) converges if 1)]([ →αGLog , as ∞→α , but, many times this circumstance does not occur. In these cases usually the idea is to normalize
)()()( ααα KfG = such that 1)]([ →αKLog , as ∞→α , and )(αf may be easily factorized in other ways. Example 2: Factorize:
(30) 41)( 2
222
++
−=αααα kG
Equation (29) does not apply directly. However, it is possible to overcome the problem
by introducing 22)( αα −= kf and 41)( 2
2
++
=αααK : it follows that
(31) )()( ααα ++ −= KkG
where through (29), or by inspection, we have:j
jK2
)(++
=+ ααα .
The condition 1)]([ →αGLog , as ∞→α may be weakened if )(αG behaves at least
as pG αα →)( as ∞→α ([3], p. 42). For instance, if )(αG is even eq.29 holds again if the integral may be considered as a Cauchy principal value integral. (32)
z
17
Many variants of the factorization formulas can be found in the literature, see for example [8, p.113]. Here only the most significant one, related to the derivative logarithmic decomposition [4, p. 23] has been reported. The derivative of the eq.25 yields
(33) )()()()(')]]([[ αα
ααα +− +== tt
GGGLogD
where:
(34) )()( αψα
α ±± =ddt
This leads to the following formula:
(35) duduG
Gjc
')')('(
)'('2
1)( ∫ ∫ −=+
α
γα
ααα
παψ )()( αψα +=+ eG
where c is a suitable constant. In many cases the introduction of a double integral is
balanced by the very simple decomposition of )()()()(' αα
αα
+− += ttGG
Example 3 The factorization of 22)( ατα −== kG introduces the function:
)(21
)(21)]([ 2 αατ
ατ+
++−
=−=kk
LogD
It follows that:
ααααα
αψ −=⇒−=+−
= ++ ∫ kGkdk
)(log)(2
1)(
1.3.2.2. Factorization of meromorphic functions Meromorphic functions can be factorized by using the Weirstrass theorem that factorizes the entire functions. More precisely when the entire function )(αf has an infinite number of simple zeroes ,...., 21 αα the following representation stands:
z
18
(36) ∏∞
=
−=
1
)0()0('
1)0()(n n
ff
neeff αα
α
ααα
Many times the zeroes nα behave as: (37) BnAn +=α as ∞⇒n . In these cases it is convenient to normalize )(αf with respect to the function defined by: [8, p.13]:
(38)
++−Γ
+Γ
=
+−=Γ ∏
∞
= 1
11)(�
1,
AB
A
ABe
eBAn
A
n
nABA α
αα
αγ
α
where )(αΓ and ...57721.0=γ are respectively the Euler gamma function and the Euler constant. By normalizing )(αf with )(�
, αBAΓ it yields, after some algebraic manipulations:
(39) )(� 1
1)0()( ,
1
αααα
α αBA
n
nh
BAn
eff Γ
+−
−
= ∏∞
=
with:
(40) ∑∞
=
−+=
1
11)0()0('
n n nAffh
α
By observing that ∏∞
=
+−
−
1 1
1
n
n
BAnααα
is bounded as BAn +≠∞→ αα , , the asymptotic
behavior of )(αf as ,∞→α is identical to the one of )(�, αBAΓ .
The function )(�, αBAΓ has the following asymptotic behavior as BAn +≠∞→ αα , [8,
p.13]
z
19
(41) )1()log(
)21(
, 2
)1()(� γ
ααααπ
α−−−
+−
−
+Γ≈Γ AAA
AB
BA eeA
AB
, BAn +≠∞→ αα ,
This last equation allows to study the asymptotic behavior of )(αf as ∞→α . A meromorphic function )(αG is the ratio of two entire functions. The factorization of
)(αG can be obtained by separating in the numerator and in the denominator the zeroes and the poles located in the two opposite half planes: 0]Im[ >α and 0]Im[ <α : (42) )(� )(�)( ααα +−= GGG where )(�)(� αα +− GandG are meromorphic functions having zeroes and poles respectively in the half planes 0]Im[ >α and 0]Im[ <α . It must be observed that in general )(�)(� αα +− GandG have non-algebraic behavior at infinity. To obtain algebraic behavior we may rewrite eq.42 in the form: (43) )( )()(�)(�)( )()( ααααα αα
+−+−
− == GGGeeGG ww where )(�)(,)(�)( )()( αααα αα
+−
+−− == GeGeGG ww . The entire function )(αwe (and its inverse) is free of zeroes and )(αw is chosen in order to ensure the algebraic behavior of
)()( αα +− GandG . When the zeroes have the asymptotic behavior (37), it will be seen later that : (44) αα qw =)( where q is constant. Example 4 The following example is relevant to the study of the bifurcation of a wave guide a in two wave guides b and c . In this case the required function to factorize is
(45) )](sin[)sin()sin()(
cbcbG
+=
ττττα , 22 ατ −= k 0,0 >> cb
Even though 22 ατ −= k is a two-valued function, )(αG is meromorphic since it is an even function of τ . Before factorizing it, it is important to determine the asymptotic behavior of )(αG as ∞→α . This behavior is algebraic:
(46) αα
α α
αα 1)( )( =∝+cb
cb
eeeG as ∞→α .
We rewrite:
z
20
(47) )(
)()()(
ααα
αa
cb
SSS
bcaG =
with :
(48) d
dSd ττα )sin()( = cbacbd +== ,,
The zeroes of the entire function )(αdS are given by dnα± where:
(49) 2
2
−=
dnkdnπα 0]Im[....,2,1 <= dnn α , cbacbd +== ,,
The asymptotic behavior of dnα as ∞→n is given by:
(50) dddn BnA +≈α , 0, =−= dd Bd
jA π , cbacbd +== ,, , cba AAA
111+=
Taking into account that )(αdS is even, eq. (36) yields:
(51) =
−== ∏
∞
=1
2
1)sin()sin()(n dn
d kddk
ddS
αα
ττα
=
−
+== ∏∏
∞
=
∞
=
−
11
11)sin()sin()(n dnn dn
ddndn ee
kddk
ddS α
ααα
αα
αα
ττα
∏∏∞
=
∞
=
−
−
+=
11
11)sin(n
nA
dnn
nA
dn
dd eekd
dkαα
αα
αα
Notice that when the zeroes are separated in two distinct infinite products, the
exponential factors dne αα∓
may be substituted by the simpler factors nAdeα∓
. The presence of an exponential is however always necessary to ensure the convergence of the infinite products. Taking into account (51) and (38), algebraic manipulations yield the following result (non algebraic at ∞ ) for the plus factorized functions:
(52) )(�
1
1)sin()( 0,
1
αα
αα
αdA
n
d
dnd
nAkd
dkS Γ
−
−
= ∏∞
=+ , cbacbd +== ,,
In this formula the infinite product behaves as a constant as ∞→α . So the asymptotic behavior of )(α+dS coincides with that of )(�
0, αdAΓ , expressed by (41):
z
21
(53) )1()log(
21
0, )(�)(γ
ααα
ααα−−−−
+ ≈Γ∝ ddd
d
AAAdAd eeMS , ∞→α
Eq.47 yields:
(54) )(
)()()(�
ααα
αa
cb
SSS
bcaG
+
+++ =
The asymptotic behavior of )(� α+G is given by:
(55) α
γααα
γαααγααα
α
α
ααα q
AAA
AAAAAA
eee
eeeeGaaa
cccbbb21
)1()log(21
)1()log(21)1()log(
21
)(� −
−−−−
−−−−−−−−
+ =∝ , ∞→α
where, by taking into account eq. (50)
(56) )loglog(1log11log11log1cac
babj
AAAAAAq
aaccbb
+−=−+=π
Eq.55 shows that )(� α+G and )(�)(� αα −= +− GG have exponential behavior as ∞→α . By looking at (43), we define: (57) )(�)(,)(�)( αααα αα
+−
+−− == GeGeGG qq with q given by eq. (56). Of course )()( αα +− GandG have algebraic behavior as ∞→α . In fact :
(58) 21
21
)(,)(−
+
−
− ∝∝ αααα GG As an numerical example let us suppose
24.2,
23.1,
21.1,2,1 λλλ
λπλ =+===== cbacbk
It is possible to evaluate the infinite product by approximating it with the finite product:
∏∏=
∞
=
−
−
≈
−
−
dN
n
dn
n
dn
ndj
ndj 11 1
1
1
1
πααα
πααα
, cbad ,,=
z
22
The choice of dN depends on how many real dnα are present. In fact, real dnα are related to modes that propagate, whereas imaginary dnα involve evanescent modes. In this case we have the following first three dnα (d=a,b,c):
,71.4,47.3,71.5 321 jaaa −=== ααα ,94.15,54.9,62.2 321 jj bbb −=−== ααα ,07.13,35.7,01.4 321 jj ccc −=−== ααα
Then two modes propagates in the waveguide a, one mode in the waveguide b and one mode in the waveguide c. The very small number of the propagating modes means that Nd can be assumed small. However, we first assumed high values of Nd by putting:
2000=== cca NNN Fig.3.5 illustrates the absolute value and the argument of )(α+G for real values of α
fig. 3.5.a: Absolute value )(α+G of the factorized function )(α+G
fig. 3.5 b: Argument [ ])(arg α+G of the factorized function )(α+G
z
23
Fig.3.6 reports the error )()()( ααα +−−= GGGe
fig.3.6: The error )()()( ααα +−−= GGGe
By assuming 20=== cca NNN we practically obtain the same plots
fig.3.7: Truncation error for 20=== cca NNN
Fig.3.7 illustrates the relative error when we change 2000=== cca NNN to
20=== cca NNN :
(59) )(
)()()( 1
ααα
α+
++ −=
GGGep
where )(α+G has been evaluated by putting 2000=== cca NNN and )(1 α+G has been evaluated by putting 20=== cca NNN . As a matter of fact )()()( ααα +−−= GGGe does not depend on q, a small error
)()()( ααα +−−= GGGe does not assure that the choice of q given by eq.(56) is correct. We ascertained the correctness of q by taking into account that the function
( )k Gα α+− and its inverse 1( ( ))k Gα α −+− must behave as a bounded not vanishing
constant as α →∞ . 1.3.2.3. Factorization of kernels involving continuous and discrete spectrum
z
24
Example 5 Factorization of:
(60) ]cos[
)(d
eGdj
ττα
τ
= , 22 ατ −= k
The asymptotic behavior of )(αG is algebraic and is given by:
(61) αα
αα
α 1)( =∝d
d
eeG as ∞→α .
The factorization of )(αG may be accomplished through the factorization of the factors τ , dje τ and ]cos[ dτ . Since we have from the factorization of τ and the decomposition (8) of τ : (62) ααατ −+=−= kkk 22
(63) ]logexp[]logexp[k
jdk
jdeee djdjdj ατπτατ
πττττ −+
== +−
it remains only the entire function ]cos[)( dc τα = to be factorized. According to eq.(36), first we must obtain all of its zeroes nα± . These are given by:
(64) ,.......2,1,0)cos( )2/1(22 ==⇒=− − ndk ndnn ααα
with :
22
22
)()2/1(
−
−=→
−=
dnk
dnk nnd
παπα
and, as ∞→n , they assume the values:
(65) BAnnAdndn +=−≈= − )2/1()2/1(αα , ABd
jAA d 21, −=−==
π , 21
−=AB
From (36), taking into account that )(αc is even, one obtains:
(66) =
−== ∏
∞
=1
2
1)cos()cos()(n n
kddcαατα
∏∏∞
=
∞
=
−
−
+=
11
11)cos(n
nA
nn
nA
n
eekdαα
αα
αα
z
25
Notice that the presence of the factors nAeα∓
it is necessary to ensure the convergence of the infinite products. Again, by normalizing with )(�
, αBAΓ , it yields the non-algebraic plus factorized functions:
(67) )(�
1
1)cos()(� ,
1
αα
αα
α BAn
n
BnA
kdc Γ
+−
−
= ∏∞
=+
(68) )(�
]logexp[)(�
αα
ατπτ
α+
+−
−
=ck
kjd
G
By taking into account that in our case as ∞→α (41),(14).
=∝
−
+Γ≈Γ≈
−−−−−−+−
+
)1()log()1()log()
21(
, 2
)1()(�)(�
γααα
γαααα
παα AAAAAA
AB
BA eeeeA
AB
c
)]1()2[log()2log()]1()2[log()2log( γπ
παα
πα
γααα
−+−−−−+−−== kd
jdjk
djkA
AkA eeee
]2logexp[k
dje dj απατ −
∝+
it follows:
α
αα qeG 2/1
1)(� ∝+
with the constant q given by
(69)
−+
−= γπ
π12log
kdjdjq
and the logarithm assumed in its principal part. The final formulas are:
z
26
(70a) ∏∞
=+
−
+−
Γ−
−−
=1, 1
1
)(�)cos(
]logexp[)(
n
n
BA
BnA
kdk
qk
jd
G
αα
α
αα
αατπτ
α
or (70b)
∏
∏
=
∞
=+
−
+−
+Γ−
−−
++−Γ
≈
−
+−
+Γ−
−−
++−Γ
=
bN
n
n
A
n
n
A
BnA
ABekdk
qk
jdAB
A
BnA
ABekdk
qk
jdAB
AG
1
1
1
1
1)cos(
]logexp[1
1
1
1)cos(
]logexp[1)(
αα
α
α
αατπτα
αα
α
α
αατπτα
α
αγ
αγ
(71) )()( αα −= +− GG
By assuming 2
1.1),101(2,1 6 λλπλ =−== − bjk and a truncated product with
200=bN , we obtain an error )()()( ααα GGG −+− that is vanishing in the range 100 100α− ≤ ≤ .
As a matter of fact )()()( ααα +−−= GGGe does not depend on q, a small error
)()()( ααα +−−= GGGe does not assure that the choice of q given by eq.(69) is correct. We ascertained the correctness of q by taking into account that the function
( )k Gα α+− and its inverse 1( ( ))k Gα α −+− must behave as a bounded not vanishing
constant as α →∞ . Example 6 Factorization of:
sin[ ]( ) j d dG ed
τ τατ
−=
Taking into account the previous consideration we obtain (miste7.nb)
z
27
1
sin( ) 1exp[ log ] 1( )
1 1
A
n
n
kd d j Bq ekd k AG
BA A A n B
αγ ατ τ α ααπα
α α
∞
+=
− −− + Γ + =
Γ − + + − +
∏
where:
, 0A j Bdπ
= − =
−+
−= γπ
π12log
kdjdjq (see eq.69)
22
nnkdπα = −
1, 2...., Im[ ] 0nn α= < ,
By assuming 621, (1 10 ), 1.12
k j dπ λλλ
−= = − = and a truncated product with
200=bN , we obtain an error )()()( ααα GGG −+− that is vanishing in the range 100 100α− ≤ ≤ .
As a matter of fact )()()( ααα +−−= GGGe does not depend on q, a small error
)()()( ααα +−−= GGGe does not assure that the choice of q given by eq.(69) is correct. We ascertained the correctness of q by taking into account that the function
( )k Gα α+− and its inverse 1( ( ))k Gα α −+− must behave as a bounded not vanishing
constant as α →∞ .
1
D:\Users\vdaniele\libri\WHnew\chapter1-4.doc 1.4. Decomposition Equations in the w-Plane. The fundamental formula to decompose a function is based on the Cauchy equations (1) and (2) of (1.3.1) :
(1) ''
)'(2
1)(1
∫ −=+ γ
ααα
απ
α dFj
F ''
)'(2
1)(2
∫ −−=− γ
ααα
απ
α dFj
F
When the function to be decomposed )(αF presents only branch points due to the
function 22 αχ −= k , it is sometimes convenient to use the previous formula in the w-plane defined in sect. 1.2.12.2. 1.4.1. Evaluation of the Plus Functions
fig.4.1: Deformation of the integration path in the α’-plane
By starting from ''
)'(2
1)(1
∫ −=+ γ
ααα
απ
α dFj
F , it is possible to first deform the line 1γ into
the line oγ (fig.4.1) surrounding the standard branch line relevant to the branch point k+=α . It yields: (2)
∑∫∫∑∫ −−
−+
−=
−−
−=+
n n
n
n n
n RdF
jdF
jR
dFj
Fowoo αα
ααα
απ
ααα
απαα
ααα
απ
αγγγ
''
)'(2
1''
)'(2
1''
)'(2
1)(1
2
where nα are the poles of )(αF (that we suppose simple for sake of simplicity) located in the lower half plane, 0]Im[ ≤α , and nR the corresponding residues. In the w-plane,
wk cos−=α , by supposing that w belongs to the proper image of α (fig.2.5 of 1.2.12.2), we have:
∫∫ −−=
− 22
'cos'cos
'sin)'(ˆ
21'
')'(
21
oo
dwww
wwFj
dFj γγ π
ααα
απ
,
(3)
∫∫∫ −−−
=−
−=− 211
'cos'cos
'sin)'2(ˆ
21'
cos'cos'sin)'(ˆ
21'
')'(
21
ooo
dwww
wwFj
dwww
wwFj
dFj γγγ
πππ
ααα
απ
where the notation )(ˆ)cos( wFwkF =− has been used. By deforming 2oγ on 3γ , defined by 0]Im[ ,]Im[ <+−= wwjw π , (fig.4.2), the poles (located in a proper region of the quadrant 4) between these two lines do contribute with
opposite sign with respect to ∑ −−
n n
nRαα
, and so we obtain:
∑∫ −−
−−−−
−=+' '3
'cos'cos
'sin)]'2(ˆ)'(ˆ[2
1)(n n
nRdwww
wwFwFj
Fαα
ππ
αγ
where the apex in the index of the sum implies that the poles to be considered are those in the lower half-plane, 0]Im[ <α , located outside the sub-region of the quadrant 4 between the lower lip of the standard branch line and the half line from k to ∞ ; this half line is defined by ∞≤≤= uuk 0with,coshα . By looking at the w-plane, it is possible to see that the poles to consider are located in the proper image of the half-plane 0]Im[ ≤α and these must have:
0]Re[ <<− wπ By introducing juw +−= π' , it yields the following expression:
∫∫
∫∫∞∞
∞
−
−−−+−−=
+−−−+−
−=
=−+−
+−−−−+−−=
−
00
0
cosh
sinh)](ˆ)(ˆ[21
coscoshsinh)](ˆ)(ˆ[
21
cos)cos()sin()](ˆ)(ˆ[
21'
')'(
21
3
du
ku
ujuFjuFj
duwu
ujuFjuFj
jduwju
jujuFjuFj
dFj
αππ
πππ
π
ππππ
πα
ααα
π γ
i.e.
3
(4) ∫ ∑
∫ ∑∞
∞
+
−−
−
−−−+−−=
=−
−+
−−−+−−=
0' '
0' '
cosh
sinh)](ˆ)(ˆ[2
1
coscoshsinh)](ˆ)(ˆ[
21)(
n n
n
n n
n
Rdu
ku
ujuFjuFj
Rdu
wuujuFjuF
jF
αααππ
π
ααππ
πα
It must be remembered that, looking at the w-plane, the poles located in the proper image of the half-plane 0]Im[ ≤α having π−<]Re[w have not been considered. 1.4.1.1. Observations In the planew − the poles to be considered are located in the proper image of the half –plane 0]Im[ <α , which does not belong to the region between 2oγ and 3γ . (fig.4.2)
fig.4.2: Deformation of the integration path in the w-plane
In the w-plane the representation (4) stands if π−>]Re[w . If the integral has been evaluated in closed form, according to the analytical continuation this closed form represents the plus function also when π−<]Re[w . Conversely, in the case a numerical integration has been adopted, the jump π−>]Re[w , π−<]Re[w has to be studied according to the observations considered in 1.3.1. The singularity u=0 present in )(ˆ juF ±−π is eliminated by the function usinh . Example
Decomposition of )(ˆsin11 wF
wk=−=
χ.
The last integral of the previous formula provides the results:
4
∫∞
+ =+
−=0 sin
1)cos(cosh
11)(ˆw
wk
duwujkj
wFππ
that is in according with the expression 1.11a of sect 1.3.1 The final formula:
∑∫ +−
+−−−+−
−=∞
+n n
n
wkR
duwu
ujuFjuFj
wFα
πππ coscoscosh
)sinh)](ˆ)(ˆ[2
1)(ˆ0
shows that the plus functions are even functions in the w- plane, regular in the point w=0 corresponding to the branch point k−=α . This is a fundamental property, that will be reconsidered and exploited in the other two parts of this work, due to the choice of π=a in eq.(72) of 1.2.12.2. Other choices of the parameter a yield less convenient properties of plus-functions in the w-plane. 1.4.2. Evaluation of the Minus Functions
fig.4.3: Deformation of the integration path γ2 in the α’-plane
Minus functions may be considered as plus functions evaluated in αα −= . By taking into account the property (78) of 1.2.12.2, παα −−=− )()( ww , it follows that the minus function )( α−+F has the form )(ˆ π−−+ wF where )(ˆ wF+ is even in w and regular as
0=w . Furthermore, the images in the w-plane of the minus functions have the property of being invariant for the substitution of w with w−− π2 , i.e.
))2((ˆ)(ˆ)(ˆ ππππ −−−−=+=−− +++ wFwFwF .
Starting from ''
)'(2
1)(2
∫ −−=− γ
ααα
απ
α dFj
F , first the integration path 2γ has to be
deformed into the line −γ (fig.4.3) surrounding the standard branch line relevant to the branch point k−=α . It yields:
(5) ∑∫∫ −−
−=
−−=
−− αα
ααα
απ
ααα
απ
αγγ
n
nSdFj
dFj
F ''
)'(2
1''
)'(2
1)(2
5
where nα are the poles of )(αF located in the proper upper half plane 0]Im[ ≥α and
nS the corresponding residues.
fig.4.4: Deformation of the integration path b_ (image of γ_ ) in the w-plane
Let’s consider now the w-plane: (6) wo cosτα −= In this plane, the line −γ has as image the line ∪ −−− = 21 bbb , and it separates the image of the proper quadrant 2 denoted by the number 2 encapsulated in a square, 5, from the image of the improper quadrant 2 denoted by the number 2, as depicted in fig.4.4 (see also fig.2.5 of 1.2.12.2). The reported versus takes into account that, running on −γ on the 'α -plane , the proper sheet lies on the left side. Let’s decompose now −b in −1b (located in Im[w]>0) and −2b (located in Im[w]<0). By introducing
)'(ˆ)'( wFF =α ''sin' dwwd oτα = it yields:
''sincos'cos)'(ˆ
21''sin
cos'cos)'(ˆ
21'
')'(
21
21
dwwww
wFj
dwwww
wFj
dFj bb ∫∫∫
−−− −−
−−=
− ππα
ααα
π γ
The substitution of 'w with 'w− in the second integral implies 12 −− −= bb . Consequently:
''sincos'cos
)'(ˆ)'(ˆ
21'
')'(
21
1
dwwwwwFwF
jdF
j b∫∫−− −
−−−=
− πα
ααα
π γ
6
By deforming −1b on the imaginary axis 0]'Im[ ,]'Im[ >= wwjw , the poles, located in a proper region of the quadrant 2, between these two lines do contribute with opposite sign
with respect to ∑ −−
n n
nSαα
. By introducing juw =' ⇒ duudww sinh''sin −= , it yields
the final results in eq.(5):
(7) ∑∫∫ −−
−−−
=−
−==∞
−− ααπα
ααα
πα
γn
nSduuwujuFjuF
jdF
jwFF
0sinh
coscosh)(ˆ)(ˆ
21'
')'(
21)(ˆ)(
2
where the poles to be considered for the computation are those located in the half-plane
0]Im[ >α located outside the sub-region of the quadrant 2 between the upper lip of the standard branch line and the half line from k− to ∞− defined by
∞≤≤−= uuk 0,coshα . For what concerns the w-plane, the poles to be considered are located in the proper image and present 0]Re[ <<− wπ (fig.4.4). In the w-plane the representation (7) stands if 0]Re[ <w . If the integral has been evaluated in closed form, according to the analytical continuation, this closed form represents the minus function also for 0]Re[ >w . Conversely, in the case a numerical integration has been adopted, the jump 0]Re[ <w , 0]Re[ >w has to be studied according to the observations considered in 1.3.1. It can be ascertained that )(ˆ wF− is a minus function since it is regular in π−=w ( oτα = )
and it results that )(ˆ)2(ˆ wFwF −=−− −− π . In this case too, as in the previously considered one, the singularity in 0=u , due to the integrand denominator wu coscosh − is de facto eliminated by the presence of the function usinh . However, sometimes, 0=u is a singular point also for the presence of a singularity in the numerator
)(ˆ)(ˆ juFjuF −− . In this case there is a multiple pole at 0=u that cannot be eliminated by the presence of usinh , and this means that the original integral on −b does not converge for 0'=w , and, to perform the integral, −b has to be hooked at 0'=w so that the hook is located in the proper region 2, in fig.4.4 encapsulated by the square 5 (fig.4.5).
7
Example
Factorization of the function 22 ]]arccos[cos[(o
oo τη
πττξ −
Φ−−= .
The factorization of this function is a key point for solving diffraction problems involving wedges with aperture angle )(2 Φ−π (Part 3). By introducing the planew − , defined by the following mapping:
)cos(woτη −= it yields:
wo πτξ Φ
−= sin
The factorization can be accomplished through the logarithmic decomposition of )(ηg
)sin(sin
)cos(
)sin(
)cos(loglog)(
ww
w
w
w
dwd
wdd
dwd
ddg
o
π
πτπ
π
ππη
ξη
ξη
ηΦ
ΦΦ
−=Φ
ΦΦ
−===
The presence of the double pole at 0=w requires an hooking of the line −b (fig.4.5). The line with the hook will be denoted by −γ . Eq.2 of 1.3.1 provides the minus part of )(ηg in the form:
'cos'cos
1
)'sin(
)'cos(
21'
cos'cos'sin)'(ˆ
21'
')'(
21)(
2
wdwww
w
jwd
wwwwg
jdg
jg
o∫∫∫
−− −Φ
ΦΦ
−=−
=−
−=− γγγ
π
πτπππ
αηα
απ
η
The contribution of the hooking is given by )0(21 R− where )0(R is the residue of the
integrand at 0'=w . It follows:
'cos'cos
1
)'sin(
)'cos(..
21)0(
21'
cos'cos1
)'sin(
)'cos(
21)(
1
wdwww
wPV
jRwd
www
w
jg
boo
∫∫− −Φ
ΦΦ
−+−=−Φ
ΦΦ
=−
π
πτππ
π
πτππ
ηγ
where V.P. means Cauchy principal value of the integral. Since the integrand function is an odd function and the integration path −b is symmetrical (fig.4.4) the V.P. integral is vanishing and it yields:
8
ηξ
ηηττη
ddd
wRg
oo
−− =
+=
−=−=
log121
cos111
21)0(
21)(
Consequently:
ητηητ
ξ +=+
= ∫− oo
d log121log
By taking into account that the factorized functions may be multiplied by a constant factor, we assume:
(8) 2
ητξ
+=−
o
1.4.3. Use of Difference Equation for Functions Decomposition The property of the function )(α+F in the w-plane allows a decomposition technique based on the use of difference equations. In fact the decomposition problem (9) )()()( ααα −+= ++ YXF may be rewritten in the w-plane in the following form: (10) )(ˆ)(ˆ)(ˆ)(ˆ)(ˆ ππ ++=−−+= ++++ wYwXwYwXwF where )(ˆ wX + and )(ˆ wY+ are even functions regular in w=0. By taking into account the properties of these function it is possible to get rid of one of the unknowns, e.g. by substituting 'w with 'w− and by subtracting it yields (11) )(ˆ)(ˆ)(ˆ)(ˆ wFwFwYwY −−=−−+ ++ ππ This equation is a difference equation in the unknown )(ˆ wY+ , and it can be solved with the method described in part 2. 1.4.4. The W-H Equation as Difference Equation. In the w-plane W-H equations may be rewritten in the form: (12) )(ˆ)(ˆ)(ˆ)(ˆ wFwYwXwG o+−−= ++ π
9
By using the previous procedure, it is possible to eliminate the unknown )(ˆ wY+ obtaining the difference equation: (13) )(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ ππππππ −−−−=−−−+−− ++ wFwFwXwGwXwG oo As it will considered in Part 3, the decomposition-factorization and the solution of difference equation are two different aspects for solving many diffraction problems. However, whereas the decomposition-factorization constitutes a closed mathematical problem, conversely difference equations may involve many solutions, and so it is necessary to take into account additional conditions for pinpointing the correct one.
1
D:\Users\vdaniele\libri\WHnew\capitolo 1.5.doc 1.5 Exact Matrix kernels factorization
1.5.1. Introduction The central problem in solving vector Wiener-Hopf equations is the factorization of a
nn× matrix. Even though this problem has been considerably studied in the past, up to now it is not known a method to factorize a general nn× matrix. A comprehensive discussion of the most recent advances, achieved in the last years, appears in ch.6 of [7]. In this section the most significant ideas for obtaining explicit matrix Wiener-Hopf factorization have been selected and reported, conversely, other procedures already relevant and presented in literature will be mentioned when considering particular cases in the next sections. Before discussing the matrix Wiener-Hopf factorization, it is interesting to mention that this problem has been considered in the more general framework of the vector Riemann-Hilbert problem too. A fundamental paper on this approach is the one by Chebotarev [22], that investigates the conditions in which the vector Riemann problem can be solved by a simple generalization of the formula solving the scalar case problem. However these conditions are so restrictive that there seems to be no hope that they could apply to practical cases. Khrapkov [23] studied the case n=2 in detail and, in particular circumstances, he stated explicit factorization formulas, by extending Chebotarev's results to an infinite line and giving the correct regularity condition at infinity too. Whereas in the Soviet Union the solution of the Riemann vector problem advanced by small but systematical steps, in western countries the interest in matrix factorization was mainly concerned with obtaining the W-H solution for the diffraction by a two impedances half plane. Even though the solution of this problem has been accomplished by the Malyuzhinets-Sommerfeld method in 1958, surprisingly the researchers did not succeed in obtaining the matrix factorization involved in the problem. Finally, in 1975 Rawlins [24] was able to solve with an ad-hoc technique the soft-hard half-plane problem. Rawlins’ work was substantially improved in 1976 when Hurd [25] introduced a new method (the Wiener-Hopf-Hilbert method) that solves a large class of Wiener-Hopf systems of order two. This method provides also the solution of the diffraction by the two impedances half plane. By reading this fundamental paper Daniele became confident that the construction of a direct factorization method should be possible for all of the problems where closed form solutions are known. After some investigations, a direct factorization formula for the problems for which the Hurd method apply has been finally obtained and published in 1978 [26]. Later on, by Luneburg and Hurd [27], it has been shown that Daniele's method essentially coincides with the results independently obtained by Khrapkov [23]. In most cases the Daniele-Khrapkov factorization formula gives inadmissible factor matrices due to the essential singular behavior of the Wiener-Hopf matrices at infinity. In another paper, [16], Daniele introduced an approach that overcomes such difficulty when the regularity conditions are not satisfied, generally yielding a non-linear equations system to solve. The factorization of Daniele-Khrapkov matrices can be rephrased as a scalar Hilbert-Riemann problem on a Riemann surface [28]. In general, with this
2
formulation, it is necessary to get rid of an essential singular behavior at infinity too. By using the Riemann surface formulation Antipov and Silvestrov [29] reduced, in some cases, the relevant non linear system to a classical Jacobi inversion problem. By introducing the Riemann theta function the Jacobian inversion problem constitutes a non linear system equivalent to the solution of a polynomial equation. The idea that allows the direct factorization for a particular class of order two matrices is based on the concept of logarithmic additive decomposition. This idea was extended by Daniele [30] and Jones [31] to arbitrary order matrices, e.g. it is possible to factorize with the logarithmic additive decomposition all the matrices commuting with polynomial matrices [30]. However, the factorization formulas involve an essential singular behavior at infinity and a cumbersome procedure is necessary to eliminate them. The Wiener-Hopf technique is the most powerful method that has been developed to solve electromagnetic fields and waves problems involving geometrical discontinuities. However, to study fields and waves in wedge regions the Sommerfeld-Malyuzhinets (S-M) method is usually preferred, especially by scientists from the former USSR, and the main reason for this preference is the belief that the Wiener-Hopf technique apply only for some particular values of aperture angles. This limiting belief has been now definitively overcome since Daniele in some of his most recent works showed that the diffraction by an impenetrable wedge having arbitrary aperture angle always reduces to a standard matrix factorization. Furthermore the W-H formulation of the wedge problem is important since it solves new problems in addition to those solved by the S-M method, e.g. all the problems defined in arbitrary angular regions can be formulated in the framework of the W-H technique (part 2) yielding Generalized Wiener-Hopf equations. In order to conclude this introduction, it should be remarked that a general formula for factoring matrices does not exist. Nevertheless there are many classes of matrices that we are able to factorize, even if it is difficult to recognize whether a given matrix belongs to one of these particular classes. In fact suitable algebraic manipulations may reduce the factorization of a given matrix to the factorization of a matrix having different nature, e.g. the entries of the matrices may be deeply modified by pre- or post multiplication with rational matrices. By taking into account the concept of weak factorization (sect.1.2.8), these operations may be admitted since the explicit factorization of the modified matrices ensures the explicit factorization of the original ones. The ability to reduce, when possible, a given matrix to a matrix that it is possible to explicitly factorize requires experience and mathematical ability (see for example [26] and [32]). 1.5.2. Factorization of triangular matrices A class of matrix kernels that we are able to explicitly factorize is the triangular matrices one. A matrix )(αG is a triangular matrix if its entries )(αijg satisfy the conditions: (1a) jiforgij <= 0)(α lower Triangular matrices or (1b) jiforgij >= 0)(α upper Triangular matrices
3
The triangular matrices can be factorized with the following method that, without any restriction and for sake of simplicity, will be detailed for upper triangular matrices only, i.e.:
(2)
nn
n
n
n
g
gggggggg
G
0000............
...00..0
.......
)( 333
22322
11211
=α
The simplest factorization of this class is the one of the order-two matrix 10
)(1 αg. The
additive decomposition of )()()( ααα +− += ggg yields the explicit factorization:
(3) 10
)(110
)(110
)(1 ααα +−=ggg
This is remarkable since the factorization problem is reduced to an additive decomposition that can always be accomplished. To factorize matrices of order two, in the general case, it is very useful the equation:
(4) +−
+−
+−
−−==22
111
22121
11
22
11
22
1211
00
10)()(1
00
0)(
ggggg
gg
ggg
G α
where the minus and plus functions follow from the factorization of the scalars
2211 gandg : (5) +−= 111111 ggg , +−= 222222 ggg Applying eq.3 to the middle matrix of the rightmost member of (4) it yields the factorization :
(6) +−
++−−+− ===
22
11
22
11
22
1211
00
101
101
00
0)()()(
gggg
gg
ggg
GGG eeααα
where:
(7) 10
10
0)(
22
11 −−−
−
= egg
gG α ,
+
+++ =
22
11
00
101
)(g
ggG eα
4
and −eg and +eg follow by the additive decomposition of
+−−
+−
− +== eee gggggg 12212
111 )()(
For what concerns arbitrary matrices of order mkn += , it is always possible to use a partition of a given triangular matrix )(αG in the form: (8)
mmkm
mkkk
mmkm
mmmkkkkk
mmkm
mkkk
mmkm
mkkk
ggggg
gg
ggg
G,22,
,,11
,,
,1
22,12,1
11,
,22,
,,11
,22,
,12,11
)()0()0()(
)1()0()()()()1(
)()0()0()(
)()0()()(
)(+
+−
+−
−
−
−==α
where the square submatrices mmkk gandg ,22,11 )()( , respectively of order k and m, are upper triangular matrices that have been factorized in the form:
kkkkkk ggg ,11,11,11 )()()( +−= , mmmmmm ggg ,22,22,22 )()()( +−= Again the factorization of the middle matrix of the rightmost member follows from the decomposition of the matrix :
mkemkemmmkkkmke gggggg ,,,1
22,12,1
11, )()()()()()( +−−
+−
− +== This yields the factorization )()()( ααα +−= GGG
mmkm
mkekk
mmkm
mkkk gg
gG
,,
,,
,22,
,,11
)1()0()()1(
)()0()0()(
)( −
−
−− =α
(9)
mmkm
mkkk
mmkm
mkekk
ggg
G,22,
,,11
,,
,,
)()0()0()(
)1()0()()1(
)(+
+++ =α
In general, the factorization of the square submatrices mmkk gandg ,22,11 )()( can be accomplished by a partition procedure if these submatrices have order one or two. Example 1: Factorization of a meromorphic triangular matrix Eq.(8) requires the factorization of scalars and the factorization of the triangular matrix
10)(1 αa
M = . The case where the function )(αa is a meromorphic function can be
tackled by the Mittag-Leffler expansion (1.3.15). For instance, if )(αa is proper ( 0)( →αa , as ∞→α ) and involves only the simple poles kα ( ,.....2,1=k ) with residues )( kR α in the lower half-plane and the simple poles kβ ( ,.....2,1=k ) with residues )( kR β in the upper half-plane, the following Mittag Leffler expansion stands:
5
∑∑∞
=
∞
= −+
−=
11
)()()(
k k
k
k k
k RRa
ααα
βαβ
α
It yields the following factorized matrices of +−= MMM
∏∞
=− −=
1 10
)(1
kk
kRM βα
β, ∏
∞
=+ −=
1 10
)(1
kk
kRM αα
α
Example 2 Factorization of the matrix
=
− 0)(
)( Lj
Lj
eeG
Gα
ααα
The previous matrix is very important in the solution of the modified W-H equations of
order two. By multiplying it by the matrix 0110
, it yields the triangular matrix :
=
−
Lj
Lj
eGe
Gα
α
αα
)(0
)(
This matrix is triangular but even though Lje α− ( Lje α ) is a minus (plus), its inverse becomes Lje α ( Lje α ), i.e. a plus (minus) function. So, the previous procedure cannot be used to obtain factorized matrices in closed form. A suitable rational approximation of
Lje α− allows us to do not overcome this difficulty. It is possible to obtain efficient approximate factorized matrices by reducing this problem to Fredholm equations. 1.5.3. Reduction of Arbitrary Matrices to Triangular Matrices The possibility of factorizing triangular matrices has spurred many attempts to reduce arbitrary matrices to triangular matrices, but, even though many times this reduction is possible, no general method exists. Furthermore, the reduction of a given matrix to a triangular matrix is, when possible, an uncommon matter of mathematical skill. From a general point of view, the best it is possible to obtain is from now on reported. First, an arbitrary matrix may be rewritten in the form:
(10) 10
10
0101 12
111
121
112122
111
11212221
1211 gggggg
ggggg
ggG
−
−− −==
where, in general, 11g , 12g , 21g , 22g are submatrices.
6
By substituting ,11121 +−− +== lllgg +−= 111111 ggg ,12
1112122 +−− ==− dddgggg and
,121
11 +−− +== hhhgg it yields the following factorization of the triangular and the
diagonal matrices present in eq.10:
101
101
101
11121 +−− =
llgg,
101
101
101 12
111 +−−
=hhgg
+
+
−
−− =
− dg
dg
ggggg
00
00
00 1111
121
112122
11
The product of a triangular matrix times a diagonal matrix is a triangular matrix that it is possible to factorize by the previous equation:
===−+
−−−
−
−−+
−
−
−
+ 101
000
00
101
111
11
11
1111
glddg
dglg
dg
l
1}{01
1}{01
00
111
111
11
+−+−−−−+
−−−
−=gldgldd
g
===+
+−+−+
+
−++−
+
+
dgdhg
dhggh
dg
00
101
0101
00 11
111111111
+
++−+−+−
−+−+=
dgdhgdhg
00
10}{1
10}{1 11
111
111
where −+,{....} denotes the plus and minus decomposed of {....} It yields: (11)
+
++−+−+−
−+−+
+−+−−−−+
−−−
−=d
gdhgdhggldgldd
gG
00
10}{1
10}{1
1}{01
1}{01
00 11
111
111
111
111
11
The previous equations reduce the factorization of G to the factorization of M:
(12) −
−+−++−+
−−+−+
−−
−−+−+−
−+−+
+−+−− +
==}{}{1}{
}{110
}{11}{01
11111
111
1
111
111
111 dhggldgld
dhgdhggld
M
7
Dealing with matrix factorizations, besides the classical direct factorization +−= MMM , it is also possible to define an inverse factorization −+= ii MMM . Since eq.(12) defines in the second member the inverse factorized −+ ii MandM of M, the following problem arises: Is it possible to explicitly obtain the expressions of the direct factorized matrices −M and
+M from the inverse factorized −+ ii MandM ? The answer is, till now, affirmative only if the considered matrices are rational [33] or, in general, for the particular case where M is a matrix of order two having the Daniele form [32]. However, even though eq.(12) does not provide explicit factorizations valid in general, in some cases eq.(11) can be useful for factorizing matrices when the entries of G have peculiar properties, e.g. if +−
− +== lllgg 11121 is constant, it follows that
0}{ 111 =+−+−− gld and consequently the factorized matrices of )( )()( ααα +−= GGG are:
10}{1
1001
1}{01
00 1
11
111
11 −−+−+
−−+−−−
−− =
dhggldd
gG
(13)
+
++−+−+
+ = dgdhg
G0
010
}{1 111
11
The condition 1
1121−gg constant can be substituted by the less stringent condition requiring
11121−gg to be at least a polynomial, because, in this case, +−+
−− }{ 11
1 gld becomes a polynomial of α too. It means that the factorized matrix
(14) 10
}{11}{01
1}{01
00 1
11
111
111
11 −−+−+
+−+−−−−+
−−−
−− =
dhggldgldd
gG
is again a minus function. Another important case allowing an explicit factorization is when +l is a rational function of α , even if this case requires the concept of weak factorization and so it will be discussed later. Similar considerations apply if +−
− +== hhhgg 121
11 is, in general, a matrix of order two, since for matrices of order two this means that it is possible to explicitly obtain the factorized matrices if a row or a column has rational entries.
8
Corollary 1: A matrix of order two with three rational entries can be explicitly factorized. It follows from the fact that three rational entries in a matrix of order two implies that a row or a column is rational. 1.5.4. Factorization of Rational Matrices. 1.5.4.1. Introduction The factorization of rational matrices is a fundamental technique for dealing with many electrical engineering problems, e.g. this factorization provides a tool for the solution of the optimal filtering problem [34], and the impedance synthesis of n-port networks [35] among others. As it will be seen later on, the factorization of rational matrices can always be accomplished explicitly [36]. Unfortunately, in diffraction theory these matrices do not occur, so the fact that constructive factorization is possible is of little importance for solving wave problems in the presence of geometrical discontinuities. However, Padé representation allows to approximate an arbitrary function with a rational function, so approximate factorizations of arbitrary matrices can be accomplished by introducing suitable Padé approximants [37] or, more generally, rational approximants of suitable entries of the matrix (sect.1.6). Rational matrices can be factorized due to the fact that every rational matrix )(αR may be rewritten in the form:
(15) )()()(
ααα
dPR =
where )(αd is a scalar polynomial and )(αP is a matrix polynomial of the type
(16) ∑=
=l
i
iiAP
0)( αα
where iA are nn× matrices whose entries are constant complex numbers. Since the factorization of the scalar polynomial )( )()( ααα +−= ddd is straightforward, eq.15 reduces the factorization of a rational matrix )(αR to the factorization of a matrix polynomial )(αP . The fundamental result that allows the factorization of )(αP is the Smith representation (e.g. see [38]) of the polynomial matrices: (17) )()()()( αααα FDEP = In this representation )(αE and )(αF are matrix polynomials of order n with constant nonzero determinant and the matrix )(αD is diagonal:
9
(18) nrdDiagD r ,....,2,1)],([)( == αα where all the scalar polynomials )(αrd are divisible by )(1 α−rd . The factorization of )(αD is straightforward: (19) )()()( ααα +−= DDD where, since )()()( ααα rrr ddd +−= :
nrdDiagD r ,....,2,1)],([)( == −− αα , nrdDiagD r ,....,2,1)],([)( == ++ αα The factorization of )(αP follows in the form: (20) )()()( ααα +−= PPP where: (21) )()()( ααα −− = DEP )()()( ααα FDP ++ = Taking into account the property of the determinant of )(αE and )(αF , )(1 α−E and
)(1 α−F are polynomial matrices too. So, the factorization of the rational matrix )()()( ααα +−= RRR is given by:
(22) ,)()()(
αα
α−
−− =
dPR ),()()()( 111 αααα −−
−−−− = EDdR
(23) ,)()()(
αα
α+
++ =
dPR ),()()()( 111 αααα −
+−
+−+ = DFdR
Even though the Smith representation (17) has a very deep conceptual importance, since it is not straightforward, in practical cases, to obtain the three matrices )(αE , )(αD and
)(αF , in general it is better to accomplish the factorization of rational matrices using different procedures. In the following, for sake of simplicity, it will be supposed that the involved poles are simple, since the presence of multiple poles requires only a slight modification of the presented procedures. 1.5.4.2. The Matching of the Singularities The Factorization of rational matrices can be accomplished in general by using the ideas (weak factorization) indicated in 1.2.8. However if simple poles are involved the following technique provides a more simple method.
10
1.5.4.2.1 A more simple technique Let’s consider a rational matrix ( )G α and its inverse 1( )G α− that put in the form:
(1.1) ( )( )( )
AGdααα
= 1 ( )( )( )
BG ααδ α
− =
where ( ) and B( ) A α α are polynomial matrices of order n and ( ) and ( ) d α δ α are scalars. In the following we will suppose:
(1.2) 1( ) ( )0 and 0 as G Gα α α
α α
−
→ → →∞
As indicated in sect. 1.2.7 the homogeneous solutions of the W-H equation
( ) ( ) ( ), i=1,2...,n( ) i i
A X Xdα α αα + −=
where , ( )0, as iX α
αα+ − → →∞
provide to evaluate factorized matrices in the form:
1 2( ) ( ), ( ),........, ( )nG X X Xα α α α− − − −= ,
11 2( ) ( ), ( ),........, ( )nG X X Xα α α α −
+ + + += Let’s introduce the functions:
( )( ) ii
p
XF ααα α
++ =
−
where pα with Im[ ] 0pα < . In the following we will suppose that pα do not belong to the null space of ( )pG α (it happens if det[ ( )] 0pG α = and yields ( ) 0i pX α− = ). Also, we
will suppose that pα do not belong to the null space of 1( )pG α− (it happens if 1det[ ( )] 0pG α− = ) and yields ( ) 0i pX α+ = .
Thus excluding these cases, apparently it seems that the only constraint is Im[ ] 0pα < We did not study the problem of the better choice of pα . Changing the value of pα it
11
can be shown (sect.1.6.4.2) that the factorized matrices ( )G α+ and ( )G α− differ only by a constant matrix. The homogeneous equation can be rewritten in the form:
( ) ( ) ( )( ) ( ) ( ), i=1,2...,n( )
i i o i oi i
o o
X X XP F Fd
α α αα α αα α α α α
− − −+ −
−= + =
− −
or:
( ) ( ) ( ) , i=1,2...,n( ) i
s ii
o
EP F Fdα α αα α α++ = +
−
where ( )i i pE X α−= and the plus ( )iF α+ and the minus ( ) ( )
( ) i i psi
o
X XF
α αα
α α− −
+
−= =
−
are standard Laplace transforms. In the following we will put [ ] [ ] [ ]1 21,0,0,0... , 0,1,0,0... ,... 0,0,0,0...,1t t t
nE E E= = = . It provides n independent solutions of :
( ) ( ) ( ) ( ), i=1,2...,n( ) i
s ii i
o
EP F F Fdα α α αα α α++ −= + =
−
For reasons of simplicity, in the following we will suppress the subscript i and will indicate:
i=1,2,..riα zeroes of [ ]d α having Im[ ] 0iα > (minus zeroes) i=1,2,..siγ zeroes of [ ]δ α having Im[ ] 0iα < (plus zeroes)
Taking into account (1.2) and that the zeroes i=1,2,..riα of [ ]d α that have Im[ ] 0iα > induce poles in ( )F α− , it yields the representation:
r
i
( )( ) + i
p i
REF ααα α α α− =− −∑
Similarly from ( )( ) ( )( )
BF Fαα αδ α+ −=
we have the representation:
s1
i
( )( ) ( ) + ip
p i
TEF G γα αα α α γ
−+ =
− −∑
12
The previous representations introduce r+ s unknown vectors or ( )n r s+ unknown scalars. We will provide ( )n r s+ equations by forcing that from the equation
( ) ( ) ( ), i=1,2...,r( )
P F Fdα α αα + −=
it follows:
( )Residue[F ( )] =R( ) ( ) i=1,2...,r'( )i
ii i
i
P Fdα ααα α αα− = += = ( r n scalar equations⋅ )
and from the equation
( )( ) ( )( )
BF Fαα αα+ −=
∆
it follows:
( )Residue[F ( )] =T( ) ( ) i=1,2...,s'( )i
ii i
i
B Fα γγα γ γ
δ γ+ = −= = ( s n scalar equations⋅ )
Solving these ( )n r s+ yields the evaluation of the residues R( ) i=1,2...,riα and T( ) i=1,2...,siγ In the previous equations the pole we considered simple poles. A slight modification of this procedure allows to face the case where multiple poles are involved. An example of matrix of ordere three (Example (matr_rat_grado_3b.nb, matr_rat_Wie.nb)) Let’s consider the factorization of the rational matrix:
(1.3)
2
2 2
2
2
2
2 2
421 1
9( ) 1 21
1 21 1
j
G
αα α
ααα
αα α
+ + + +
= + + +
that has as inverse:
13
( )( )( )
( )( )( )
( )( )
( )( )( )
( ) ( )( )( )
( ) ( )( )
22 4 2 2 2 4 2
6 4 2 6 4 2 6 4 2
2 4 2 2 4 2 2 4 21
6 4 2 6 4 2 6
1 4 7 5 1 4 7 10 1
4 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5
1 5 2 1 4 2 2 1 5 4 9( )
4 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5 4
j j j j j j j j j
j j j jG
j j j j j j
α α α α α α α
α α α α α α α α α
α α α α α α α α αα
α α α α α α α−
+ + − + + +− −
+ − − + + + − − + + + − − + +
+ − + + + − − + + − + −= −
+ − − + + + − − + + + ( )
( )( )( )( )
( ) ( )( )
4 2
22 4 2 2 26 4
6 4 2 6 4 2 6 4 2
1 4 (3 7 ) 5
1 2 2 2 1 12 3 14 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5
j j j
j jj j j j j j j j j
α α
α α α α αα αα α α α α α α α α
− − + +
+ + − + + − + −− + − − + + + − − + + + − − + +
The exact factorization of this matrix was accomplished with the method indicated in the previous section by assuming 2p jα = − . The expressions of the exact plus factorized matrix ( )G α+ is reported in the following expression
( ) , ,G α+ = +1 +2 +3g g g where the three column matrices , and +1 +2 +3g g g are given respectively by ( ( )i j= :
14
1
D:\Users\vdaniele\libri\WHnew\chapter1.5.4.3.doc 1.5.4.3 Use of the Matrix Polynomial Resolvent to Factorize Rational Matrices
Consider the rational matrix )(αR defined by : )()()( α
αα dNR =
where )(αN is a polynomial matrix of order n and )(αd is a polynomial scalar. )(αaN is the adjoint of )(αN so defined:
)()(
)(1
αα
αa
a
dN
N =−
where )(αad the determinant of )(αN
By substituting in the previous equations, it follows:
(30) )()()()()( 1 αα
ααα −== Pd
dNR a
where the polynomial matrix )(1 α−P , of degree ! and order n, is defined by:
(31) )()()( ααα aNdP =
If the matrix coefficient nA of the power lα in )(αP is invertible, it follows that, without loss of generality )(αP may be supposed to be a monic polynomial, i.e. :
(32) )(αP = ol
ll AA +++ −
− ......1 11αα
In fact, if 0]det[ ≠nA it is convenient to deal with the rational matrix )(1 αRAn− , ensuring
the introduction of a monic polynomial.
Eq.(31) allows us to use the resolvent equation [38] valid for a monic polynomial matrix:
(33) YTXP 11 )1()( −− −= αα
where the constant matrices X, T and Y have order (n, n !), (n!,n!), (n!,n) respectively, and define a triple of the monic matrix . There are many possibilities in choosing the triple X, T and Y related to the polynomial matrix )(αP , e.g. it is possible to choose:
2
(34) 0....001=X ,
121 ...........................0000...1000...010
−−−−−
=
lo AAAA
T ,
1...000
=Y
where the submatrices are all of order n
The constant matrix T can be represented by the Jordan form:
(35) 1−= SJST
where S is the eigenvectors matrix and J is a Jordan matrix formed by Jordan blocks on
the main diagonal:
(36)
kJ
JJ
J
...00............0..00..0
2
1
=
The Jordan blocks rJ are sub matrices of order s given by triangular matrix of the type:
(37)
o
o
o
rJ
α
αα
...00............0..00..1
=
They are connected to a single eigenvalue oα of T having multiplicity s. Since 111 )1()1( −−− −=−= SJSTM αα , the factorization of 1)1( −−Ta is reduced to that of
1)1( −− Jα . It is possible to show that this matrix is triangular, so that the results of sect. 1.4.2 apply in factorizing +−= MMM . The matrices X and Y are, in general, rectangular, so it is not correct to write the factorized matrices of +−
− == GGGP )(1 α in the form:
−− = MXG , YMG ++ =
Conversely, it is possible to obtain factorized matrices by considering the homogeneous problem associate to )(αG (see sect.1.2.9 ):
3
−+ = FGF
that we can rewrite as:
−+−−−− =− FFYTTTTX llll 1111 )(α
Note that the following property holds [38, page57]:
(38) 111 =−− YTTX ll
so that it is possible to obtain:
−−−
+−−−− =− FYTTXFYTTTTX llllll 111111 )(α
or:
−−
+−−− =− FYTFYTTT llll 1111 )(α
or
−+ =NMN
where
+−
+− = FYTNT ll 11 , −
−− = FYTN l 1
The previous equations may be inverted by pre-multiplying with 1−lTX . It follows by (39) that:
+−
+ = NXTF l 2/)1(3 , −−
− = NTXF l 1
So, if )}(),({ αα −+ ii NN nli ,....2,1= is the set of homogeneous solutions of
+−− =−= MMTM 1)1(α given by:
)(,),........(),()( 21 αααα −−−− = nlNNNM , 1
21 )(,),........(),()(−
++++ = αααα nlNNNM
the homogeneous problem associated to the factorization of the matrix +−
− == GGGP )(1 α has the solutions:
+−
+ = il
i NXTF 2/)1(3 , −−
− = il
i NTXF 1
4
If it is possible to choose n independent solutions )}(),({ αα −+ rr FF , nr ,....2,1= among the nl previous ones, the factorized matrices −G and +G of G are given by:
(40) )(),........(),()( 21 αααα −−−− = nFFFG , 1
21 )(,),........(),()(−
++++ = αααα nFFFG
Generalized factorization of rational matrices.
The previous procedure can be applied to the generalized factorization 1.2.11.2 of rational matrices of α . It reduces the generalized factorization to a generalized decomposition problem, considered in section 1.2.11.1
1.5.1.1. Use of the Realization Theory to Factorize Rational Matrices
A rational matrix )(αR where DR =∞− )(1 may be represented in the form given by [36]:
(41) DBACR +−= −− 11 )1()( αα
The previous representation is typical of System Theory, where )(1 α−R is used to describe the transfer function and A, B, C and D are the matrices arising from the state equations of a considered system. For a known )(αR , the evaluation of this quadruple as well as the factorization of the realization form are reported in [36] 1.5.1.2. Use of the Logarithmic Decomposition As it happens for the scalar case, the most powerful method for factorizing matrices
)(αG is the one relying on the concept of logarithmic decomposition [39] consisting in the additive decomposition of )](log[ αG (section 1.5.5): (42) )()()](log[ αψαψα +− +=G If the two matrices )()( αψαψ +− and commute the factorization of
)( )()( ααα +−= GGG is given by the equation: (43) )](exp[)( αψα −− =G , )](exp[)( αψα ++ =G It must be observed that the application of eq.s43 requires that the matrices
)()( αψαψ +− and commute. In order to show this fact for the rational matrices let us consider the Cayley theorem:
5
(44) )()(...)()(1)(])()(log[ 1
11 ααψααψαψαα −
−+++= nno PP
dP
where n is the order of the rational matrix )()()(
ααα
dPR = . If )(αR has distinct
eigenvalues )(αλ i , the functions )(αψ i may be obtained by the Sylvester formula [40, p.156]:
(45) ∑ ∏= ≠ −
−=
n
i
n
ij ji
ji
RR
1
1)(]log[)](log[
λλλα
λα
The decomposition of (46) )()()( αψαψαψ +− += iii 1.,.1,0 −= ni yields the factorized matrices: (47) )(ˆ)(ˆ)()( αααα +− ⋅== GGRG where (48) )]()(...)()(1)(exp[)](ˆ 1
)1(1 ααψααψαψα −−−−−− +++= n
no PPG (49) )]()(...)()(1)(exp[)](ˆ 1
)1(1 ααψααψαψα −+−+++ +++= n
no PPG Since )(αP is a polynomial in the variable α , these factorized matrices are regular together with their inverses in the half-planes 0]Im[ ≤α and 0]Im[ ≥α respectively, furthermore they are both functions of the same matrix )(αP so they do commute. However, the factorization thus accomplished is non-standard and we cannot use these factorized matrices since they have non-algebraic behavior as ∞→α . In order to overcome this difficulty we observe that factorized matrices differ only for the presence of entire factors so that the factorized matrices )(α−G and )(α+G having algebraic behavior have the following form: (50) )()(ˆ)( ααα UGG −− = , )(ˆ)()( 1 ααα +
−+ = GUG
The explicit evaluation of the entire matrix )(αU is a formidable task. We will introduce the following technique for accomplishing it and, apparently, no better method seems available. By assuming well-posed problems, we are dealing with factorized matrices
)(, α+−G and their inverses whose elements behave as pα ( 1<p ) for ∞→α . Consequently, it is possible to use Mittag Leffler expansions for these matrices. By taking into account the presence of simple poles, it yields:
6
(51a) )0()0(ˆ)(11ˆ)()(ˆ)( UGUTUGG ss ss
s −−− +−
−
+== ∑ β
ββαααα
(51b) )0()0(ˆ)(11ˆ)()(ˆ)( 111 UGURUGG nn nn
n−+
−+
−+ +
−
−== ∑ α
αααααα
In these representations sβ− (s=1,2,..) and nα (n=1,2..) are the simple poles, with
residues sT̂ and nR̂ of )(ˆ α−G and )(ˆ 1 α−+G respectively. )0(U is an arbitrary constant
matrix that can be put equal to the identity matrix : 1)0( =U . From equation (51b) it follows:
(52)
+
−
−= −
++ ∑ )0()0(ˆ)(11ˆ)(ˆ)( 1 UGURGU nn nn
n αααα
αα
Since )(ˆ α+G and nR̂ are known, the previous equation allows the evaluation of the entire matrix )(αU through its samples )( nU α . To obtain a system of equations for these samples we substitute
(53)
+
+
+−−=− −
++ ∑ )0()0(ˆ)(11ˆ)(ˆ)( 1 UGURGU nn nns
nss αααβ
ββ
in the equation (51a). It yields:
(54)
+= ∑−
− )()(ˆ)()(ˆ)( 1 ααααα VURQGU nn
nn
where the known functions )(αnQ and )(αV are given by:
(55) )(ˆˆ))((
)( sss snsn
n GTQ ββαβαα
αα −++
−= +∑
(56) )0()0(ˆ)(ˆˆ11)0(ˆ)( 1 UGGTGV sss ss
−
−
++= −
++− ∑ βββα
α
By substituting in eq.54 α with mα the following system in terms of the unknowns samples )( nU α stands:
7
(57)
+= ∑−
− )()(ˆ)()(ˆ)( 1mn
nnmnnmm VURQGU ααααα
Since the number of poles involved is finite, the system can be solved in closed form. 1.5.2. The factorization problem and the functional analysis It must be observed that in all the cases it is possible to reduce the solution of the Wiener-Hopf equations and of the generalized Wiener-Hopf equations too to an integral equation, e.g. the considerations made in sections 1.2.11.1 e 1.2.11.3 yield the integral equation for the plus unknown of a classical Wiener-Hopf equation:
(58) ∫∫ −=
−
−−
+−∞
∞− +
−
+1
)()()(1)()(2
1)(11
γ αα
ααα
πα du
uuFGduuF
uuuGG
jF
The main methods for solving integral equations are the iterative method, the Fredholm determinant method and the moment method, which include the Ritz-Garlekin method and the least square method [41]. 1.5.2.1 The iterative method The iterative method is based on the Neumann equations that invert the operator A−1 : (59) ( ) 1 21 1 .....A A A−− = + + + , 1<A . Equation (59) applies to (58) by assuming :
11 ( ) ( ) 1( , )
2G G uA A u
j u uαα
π α α
− → = − − − −
A general factorization iterative algorithm that avoid to introduce an associated integral equation, is indicated in [42, 32]. By substituting for instance AGGG −== −+ 1 ,
RG +=−+ 11 , an explicit formula for R is given by:
(60) ........}}}{{{}}{{}{ +++= ++++++ AAAAAAR The practical calculation of more than one or two terms in eq.60 may be prohibitive. For instance it is very regrettable that also dealing with rational matrices it seems that the involved series cannot be evaluated in closed form. Besides in general it is very difficult to ascertain the convergence of these series. So, further studies are necessary in order to make eq.60 effective in practice.
8
1.5.2.2 The Fredholm determinant method The inversion of operators defined in finite spaces (matrices) can be accomplished by using the determinant theory. To extend this method to infinite spaces, firstly we must define in an abstract way the determinant of a arbitrary operator L . Starting by the traditional definition we obtain (Marcuvitz): det[ ] exp{ [log ]}L Tr L= with [ ]Tr M is the trace of the operator M . To invert 1L Kλ= − we put:
( ) ( )1 11 ( )1 1 1 1( )
DL K K K λλ λ λ λλ
− −− = − = + − = +∆
where λ is an arbitrary complex variable and :
( ) 1( ) 1 det(1 )D K K Kλ λ λ−= − −
( ) det(1 )Kλ λ∆ = − For K constituted by compact operators, as it happens in finite space the functions
( )D λ and ( )λ∆ are entire on λ . (this is the reason of the normalization with the determinant ( )λ∆ ) . Consequently we can expand ( )D λ and ( )λ∆ with series that are valid for arbitrary value of λ :
2
0 1 2( ) ... ( 1) ...2! !
nn
nnλ λλ λ∆ = ∆ − ∆ + ∆ + + − ∆ +
2
0 1 2( ) ... ( 1) ...2! !
nn
nD D D D Dn
λ λλ λ= − + + + − +
where:
0 1∆ = , 1 [ ]Tr K∆ = , ( )222 [ ] [ ]Tr K Tr K∆ = − , ……
0 1D = , 21 [ ]D K Tr K K= − , ( )2 2 2 3
2 [ ] [ ] 2 [ ] 2D K Tr K K Tr K K Tr K K= − − + , …
( )11n n nD K n D −= ∆ −
9
For integral operator M:
( , ) ( )M f M u f u duα⋅ → ∫ we have:
[ ] ( , )Tr M M u u du= ∫ , 2 ( , ) ( , ) ( )M f M t M t u dt f u duα ⋅ → ∫ ∫
and so on.
By assuming 1λ = , 11 ( ) ( ) 1( , )
2G G uK K u
j u uαα
π α α
− → = − − −
, the previous equations
provide the exact solutions of the W-H equations (and the involved factorization problem). However we face with an infinite series whose coefficients are multiple integrals of the type:
1 2 1 2..... ( , ) ( , ).... ( , ) .. , 1, 2,..r r
r
K t K t K t u dt dt dt rα α∞ ∞ ∞
−∞ −∞ −∞⋅ =∫ ∫ ∫!""#""$
Despite the considerable progress in the numerical evaluation of integrals, today the numerical evaluation of the previous expressions is impracticable. Conversely the numerical solution of Fredholm equations by using quadrature formulas is possible and constitutes the best method for obtaining efficient approximate solutions of the W-H equations (see section 1.6.2) 1.5.3. Improving Approximate Factorizations Approximate factorized matrices can be improved by using the following considerations. Let us suppose that the matrix )(αG is approximated by )(~ αG and )(~),(~ αα +− GG are the known factorized matrices of )(αG , i.e. . )(~ )(~)(~ ααα +−= GGG By introducing (61a) ),()(~)(~ ααα −−− ∆= GG (61b) )(~)()(~ ααα +++ ∆= GG such that it is possible to rephrase the problem in the following way:
10
(62) εααεαααα +=+== +−+− )(~)(~)(~)()()( GGGGGG it follows (63) )(~)()(~)(~)(~1)()( 1111 ααααεααα −
+−−
−+
−−+− =+=∆∆ GGGGG
Eq.63 shows that the errors in the factorized matrices )(~),(~ αα +− GG can be exactly evaluated by factorizing the matrix )(~)()(~)(~)(~1 1111 ααααεα −
+−−
−+
−− =+ GGGGG . Since the
norm of the error )(~)(~ 11 αεα −+
−−−= GGA could be assumed to be less than one, ( 1<A ),
the application of the technique previously described in section 1.4.6 yields:
Ξ+=∆−+ 11 where an explicit formula for Ξ is (64) ........}}}{{{}}{{}{ +++=Ξ ++++++ AAAAAA Dealing with good approximations, the first term { }+A provides the efficiency of the approximation. By equations reported in section 1.3.1.1, it is possible to have:
(65) { } dtt
tGttGj
GGA ∫∞
∞−
−+
−−−
+−−+ −
−−=≈Ξα
επ
ααεα)(~)()(~
21)(~)()(~
21 11
11
The first term of the second member shows that the error )(~)( ααε GG −= locally
contributes to the error Ξ . Due to the presence of the factor α−t
1 in the integral, the
third terms contributes to the local error too, provided that )(~)()(~ 11 tGttG −+
−− ε is
sufficiently vanishing as ∞→t . 1.5.4. Factorization of Meromorphic Matrix Kernels with an Infinite Number of
Poles The problem of the factorization of matrix kernels that are meromorphic with an infinite number of poles is extremely important for studying in closed form the effect of discontinuities in waveguides. The concept of the matching of the singularities described in section 1.2.9 and the method reported in section 1.5.4.2.1 apply again, but they yield to the solution of a system with an infinite number of equations. The solution of this system cannot generally be accomplished in closed form, thus approximate techniques are required. Besides the two methods described in the next sections 1.2.9 and 1.5.4.2.1, very effective are the techniques based on rational approximants of the kernel considered in sect.1.6. The solution of the infinite system of equations can be accomplished by a truncation procedure, even if it is necessary to be very careful in performing it, since no convergence results are guaranteed. In general, it is advisable to keep in mind that from a
11
mathematical point of view, the solution of an infinite system of equations is equivalent to the inversion of an operator A that sometimes must be regularized. The regularization of an operator is a well known method to obtain an approximate inversion of the operator when no closed form solutions appear to be possible. Functional analysis suggests to rephrase the operator A in the form:
oA A C= + where oA has an inverse that can be obtained in closed form and C is a compact operator. This leads to the concept of the dominant equation: (71) oA x C x y⋅ + ⋅ = or to the regularized equation: (72) yAxT o
1)1( −=+ where CAT o
1−= is a compact operator. The compactness of T legitimates the use of a truncation procedure or other approximate techniques, e.g. in factorizing meromorphic matrices often the regularization requires the inversion of an infinite matrix whose entries are
ij −− − βα1
The inversion of this matrix can be obtained in closed form [43, 44]
10
(62) εααεαααα +=+== +−+− )(~)(~)(~)()()( GGGGGG it follows (63) )(~)()(~)(~)(~1)()( 1111 ααααεααα −
+−−
−+
−−+− =+=∆∆ GGGGG
Eq.63 shows that the errors in the factorized matrices )(~),(~ αα +− GG can be exactly evaluated by factorizing the matrix )(~)()(~)(~)(~1 1111 ααααεα −
+−−
−+
−− =+ GGGGG . Since the
norm of the error )(~)(~ 11 αεα −+
−−−= GGA could be assumed to be less than one, ( 1<A ),
the application of the technique previously described in section 1.4.6 yields:
Ξ+=∆−+ 11 where an explicit formula for Ξ is (64) ........}}}{{{}}{{}{ +++=Ξ ++++++ AAAAAA Dealing with good approximations, the first term { }+A provides the efficiency of the approximation. By equations reported in section 1.3.1.1, it is possible to have:
(65) { } dtt
tGttGj
GGA ∫∞
∞−
−+
−−−
+−−+ −
−−=≈Ξα
επ
ααεα)(~)()(~
21)(~)()(~
21 11
11
The first term of the second member shows that the error )(~)( ααε GG −= locally
contributes to the error Ξ . Due to the presence of the factor α−t
1 in the integral, the
third terms contributes to the local error too, provided that )(~)()(~ 11 tGttG −+
−− ε is
sufficiently vanishing as ∞→t . 1.5.4. Factorization of Meromorphic Matrix Kernels with an Infinite Number of
Poles The problem of the factorization of matrix kernels that are meromorphic with an infinite number of poles is extremely important for studying in closed form the effect of discontinuities in waveguides. The concept of the matching of the singularities described in section 1.2.9 and the method reported in section 1.5.4.2.1 apply again, but they yield to the solution of a system with an infinite number of equations. The solution of this system cannot generally be accomplished in closed form, thus approximate techniques are required. Besides the two methods described in the next sections 1.2.9 and 1.5.4.2.1, very effective are the techniques based on rational approximants of the kernel considered in sect.1.6. The solution of the infinite system of equations can be accomplished by a truncation procedure, even if it is necessary to be very careful in performing it, since no convergence results are guaranteed. In general, it is advisable to keep in mind that, from a
11
mathematical point of view, the solution of an infinite system of equations is equivalent to the inversion of an operator A that sometimes must be regularized. The regularization of an operator is a well known method to obtain an approximate inversion of the operator when no closed form solutions appear to be possible. Functional analysis suggests to rephrase the operator A in the form: (70) A = Ao + C where Ao has an inverse that can be obtained in closed form and C is a compact operator. This leads to the concept of the dominant equation: (71) Ao x + C x= y or to the regularized equation: (72) yAxT o
1)1( −=+ where CAT o
1−= is a compact operator. The compactness of T legitimates the use of a truncation procedure or other approximate techniques, e.g. in factorizing meromorphic matrices often the regularization requires the inversion of an infinite matrix whose entries are
ij −− − βα1
The inversion of this matrix can be obtained in closed form [43, 44]
1
8/29/2007 D:\Users\vdaniele\libri\WHnew\chapter1.5.5.doc 1.5.5 Use of the Logarithmic Decomposition The method reported in section 1.5.4.5 can also be used for arbitrary meromorphic matrices. It yields again the system (57). When the number of poles involved is finite (as it happens for quasi rational matrices) the system can be solved in closed form. Conversely, an infinite number of poles leads to an infinite number of samples )( nU α . In this case it has been shown that in a suitable vector space the operator associated to the matrix of infinite order involved in the system is compact. This means that we can solve the system of infinite unknowns )( nU α by an efficient truncation procedure or iterative methods. Moreover, a suitable normalization of the unknowns allows us to avoid the evaluation of very complicated functions involved in the starting factorized matrices
)(ˆ α−G and )(ˆ α+G . More details on this technique are presented in the cited work [45].
1.5.5.1 The Concept of Weak Factorization In many cases we are able to factorize a matrix )(αG in the form: (73) )(~)(~)( ααα +−= GGG where )(~ α−G , )(~ α+G and their inverses have algebraic behavior at infinite but they are not true minus and plus functions respectively, due to the presence of a finite number of offending poles located in the respective regularity half planes. For instance if
( ) ( ) ( ) ( ) ( )G R M M Tα α α α α− += where ( )R α and ( )T α are rational we can obtain the
factorization : ( ) ( ) ( )G R Mα α α− −=! , ( ) ( ) ( )G M Tα α α+ +=! This type of factorization has been called weak factorization [7] (see also sect.1.28). If the offending poles are all simple and it is supposed that the poles of 1( )G α−
+! located
in Im[ ] 0α > and the poles of ( )G α−! located in Im[ ] 0α < are different from the
offending poles rα and rβ of )(~ α+G and 1( )G α−−! , a simple procedure consists to
rewrite eq.(73) in the form ( ( ) ( )o
RF Xα αα α− −= +
−, Im[ ] 0oα < )
1
1 11
( )( ) ( )
( ) ( )( ) ( )
s or
s r o
o s r
o s r
A GBG F R
G G A BG X R
αα αα α α β α α
α αα αα α α α α β
−−
+ +
− −− − −− −
− − − =− − −
−= + − −
− − −
∑ ∑
∑ ∑
!!
! !!
where sα are poles located in Im[ ] 0α > (offending poles of the first member) and
2
rβ are poles located in Im[ ] 0α < (offending poles of the second member) . The
quantities rA and sB are respectively the unknown residues of ( ) ( )G Fα α+ +! in rα
and of 1( ) ( )G Fα α−− −! in rβ :
+Res[G ( )] ( )rr rA Fα αα α= += ⋅!
1Res[ ( )] ( )
sr sB G Fα βα β−− = −= ⋅!
The regularity of the two members in the respective regularity half -planes yields:
1
1 11
( )( ) ( ) 0
( ) ( )( ) ( ) 0
s or
s r o
o s r
o s r
A GBG F R
G G A BG X R
αα αα α α β α α
α αα αα α α α α β
−−
+ +
− −− − −− −
− − − =− − −
−+ − − =
− − −
∑ ∑
∑ ∑
!!
! !!
It follows that:
11
1 1
( )( ) ( )
( ) ( )( ) ( )
s or
s r o
o s r
o s r
A GBF G R
G G A BX G R
αα αα α α β α α
α αα αα α α α α β
−− −
+ +
− −− −
− −
= + + − − −
−= − − − − − −
∑ ∑
∑ ∑
!!
! !!
The regularity of ( )F α+ in the poles of 1( )G α−
+! located in Im[ ] 0α > and that of
( )X α− in the poles of ( )G α−! located in Im[ ] 0α < introduce the equations that allow
to evaluate the unknowns sA and rB . A slight modification of this procedure is necessary if the offending poles are multiple, but this case has in practice very little importance and we will not pursue it in this discussion. Sometimes this method complicates when all the elements of 1( )G α−
+! and ( )G α−
! have the same offending pole located respectively in Im[ ] 0α > and Im[ ] 0α < . In fact, in this case null space not vanishing arise that introduce additional unknowns. A different but less used procedure introduces an entire functions. First it is possible to get rid of the offending poles, by defining by )()( αα ++ nandd the polynomials having their zeros in the offending poles of )(~ α−G and )(~ 1 α−
−G . Similarly )()( αα −− nandd are the polynomials having their zeros in the offending poles of )(~ α+G and )(~ 1 α−
+G . The homogeneous W-H equation, relevant to eq.1, can be so rewritten in the form: (74) )()(~)()(~ 1 αααα −
−−++ = FGFG
3
By multiplying by )()( αα +− nd it yields:
)()()(~)()()()(~)()( 1 ααααααααα wFGndFGnd == −−−+−+++−
where, according to the considerations of section 1.3.4, )(αw is a polynomial in α . It follows that:
(75) )()()()(~
)(1
αααα
α+−
−+
+ =ndwGF ,
)()()()(~
)(αααα
α+−
−− =
ndwGF
So, in order to have true plus and minus functions )(),( αα −+ FF , the following conditions must be satisfied: (76a) 0)()(~ 1 =−−
−+ kk wG αα where −kα are the zeroes of )()( αα −− nd
(76b) 0)()(~ 1 =++
−− kk wG αα where +kα are the zeroes of )()( αα ++ nd
Eq.s76 allow us to obtain the polynomials )(αw as a combination of the null spaces of the matrices )(~ 1
−−+ kG α and )(~ 1
+−− kG α . In practical situations in which the problem of the
factorization of )(αG is well posed we have n and only n independent polynomials )(αw that are solutions of eq.s76 so that n homogeneous solutions of eq.74 allow us to obtain the true factorized matrices according to sect.1.2.9. Later on will be considered in detail some examples of the use of the weak factorization . The weak factorization may be generalized to deal with the very important cases in which there are infinite offending zeroes [14,46-48], e.g. the factorization of an arbitrary meromorphic matrix can be considered as a weak factorization problem. In these cases the problem of truncating effectively an infinite number of equations must be considered very carefully 1.5.5.1.1 An Example The problem concerning the diffraction by two parallel opposite half-planes yields the following W-H equation:
1 12
2 2
2101
2
j d
oj d
j d p
e X X FkX Xee k
τ
ττ
τ
α ατ
−
+ −−
− + −
− ⋅ = +
−−
4
where 2 2( ) kτ τ α α= = − Since the plus and minus unknowns are Laplace transform, the previous equations show that as α → ∞ we have the asymptotic behaviors :
3/ 21X α −+ ≈ , 1/ 2
2X α −+ ≈ , 1/ 2
1X α −− ≈ , 3/ 2
2X α −+ ≈
The following relationship holds: ([46], [7, p.302])(semipiani_invertiti00.nb):
2
2 1 ( ) ( ) ( )( )1 ( ) 1 0( )1 ( )( ) ( ) 02
j d
j dj d
f g fe fk fgee k fg f
τ
ττ
τ α α ααα
ααα α
τ
−+ +
−−−
−+−
− − − = ⋅ − − where (see eq.(48), p.54): ( ) ( )dg jkd Sα α=
log logd j d jj d k ke e e
τ τ α τ τ ατ π π
+ −− −− = ,
log( ) , ( ) ( )
d jkf e f f
τ τ απα α α
−−
+ − += = − By observing that:
( ) ( ) ( ) ( )q qd d dg jkd S jkd S e S eα αα α α α− +
− += = where ( )dS α± are defined by equations (52), p.55) and q is given by (eq.69,p.60) we will use the (not algebraic) factorized functions:
( ) ( ) ( )g g gα α α− += with:
( ) ( ) , ( ) ( )qdg jkd S e g gαα α α α+
+ + − += = − . It is important to observe that as α → ∞ :
5
1/ 2( ) ( )f gα α α −+ + ≈
We observe that 2
2
( ) ( ) ( )1
2
j d
j dj d
ekG G G
ee k
τ
ττ
τ
α α α
τ
−
− +−−
− = = ⋅
−
can be weakly factorized in the form:
1( )1 ( )( )( )
( ) ( ) 0
ffG
gg f
ααα
αα α
−−−
−−
= −
( ) ( ) ( )
1( ) ,1 0( )( )
f g fG
gf
α α αα
αα
+ +
++
+
− − = ⋅
The factorized matrices are weakly since their inverses presents infinite offending poles. In fact we observe that:
1
101 ( )( )( )
( ) ( ) ( )fG
gg f f
ααα
α α α
−−−
+− −
− = −
1
0 ( ) ( )1( ) ,1 ( )( )
( )
g fG
fgf
α αα
ααα
+−+
+−+
+ = ⋅ − −
Following the general procedure described previously we have:
( ) ( ) 1,( ) ( )( ) ( )
d dn gn g
α αα αα α
+ −
+ +
− −
= ===
6
By using the W-H technique, it yields the two equations:
1 2 2 11( ) ( ) ( ) ( ) ( ) ( ) ( )( )
f X f g X X wf
α α α α α α αα+ + + + −
−
+ = =
1
1 2 2
( ) ( )1 ( ) ( )( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )
p po
p
p po
p
g fX g F
fg f g f
g f X f X g F w
α αα α
α α α
α α α αα α α α α α α
α α
− −+ +
+
− − − −− − − − +
− =−
−= + + =
−
where 1( )w α and 2 ( )w α are entire functions. Let’s consider the meromorphic functions:
11
( )( )( )
wrg
ααα−
−
= , 22
( )( )( )
wrg
ααα+
+
=
From the two previous equations we obtain:
12 1 1
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )
wX f w f g f g rg
αα α α α α α α αα− − − − − − −
−
= = =
Taking into account the asymptotic behaviors of 3/ 2
2 ( )X α α −− ≈ and
1/ 2( ) ( )f gα α α −− − ≈ , it follows that 1 ( )r α− is vanishing as α → ∞ . It yields the Mittag-
Leffler representation:
111
1
( )( ) 1( )( ) '( )
dn
n dn dn
wwrg g
αααα α α α
∞
−=− −
−= =
− +∑
where ( eq.49,p.54) 2 2( )dn
nkdπα = −
Taking into account that:
1 2
2
( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( ) ( )
p po
p
p po
p
g fX f g F f w
g ff g F f g r
α αα α α α α
α α
α αα α α α α
α α
− −+ + + +
− −+ + + + +
= + =−
= +−
in similar manner ( 3/ 2
1 ( )X α α −+ ≈ and 1/ 2( ) ( )f gα α α −
+ + ≈ ):
7
22
21
( )( ) 1( )( ) '( )
dn
n dn dn
wwrg g
αααα α α α
∞
+=+ +
= =−∑
The solution of the W-H equations provide the other unknowns 1/ 2
2 ( )X α α −+ ≈ and
1/ 21 ( )X α α −− ≈ through the equations:
12 2
( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )
p po
p
g fr fX r Ff g f g
α αα αα αα α α α α α
− −−+ +
+ + − −
= − +
−
1 2
1
( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )( )
p p oo
p p
g f Ff r rX Ff g f g f g
α αα α ααα α α α α α α α α α
− −− +−
+ + − − − −
= − + + −− −
Now in these unknowns functions there are present offending poles. Eliminating these poles allows to evaluate the two meromorphic functions 1 ( )r α− and 2 ( )r α+ . Taking into account that in the second equations the offending pole pα is only apparent, we have the offending poles , 1, 2,...dn nα− = ∞ for the first equation and , 1, 2,...dn nα+ = ∞ for the second equation. In the first equation the residue in , 1, 2,...dm mα− = ∞ must be vanishing. It yields:
12
( ) ( )Res[ ( )] ( ) ( ) 0( ) ( ) ( ) '( )
p pdm dmdm o
dm dm dm dm dm p
g fr f r Ff g f g
α αα α αα α α α α α
− −−+
+ + − −
− −− − + =
− − − − − −
Similarly in the second equation the residue in , 1, 2,...dm mα = ∞ must be vanishing. It yields:
1 2( ) ( ) Res[ ( )] 0( ) '( ) ( ) ( )
dm dm dm
dm dm dm dm
f r rf g f g
α α αα α α α
− +
+ + − −
− + =
Now:
11
( )Res[ ( )]'( )
dmdm m
dm
wr bg
ααα−
−
−− = =
−,
8
22
( )Res[ ( )]'( )
dmdm m
dm
wr ag
ααα+
+
= =
22
1
( ) 1( )'( )
dndm
n dn dm dn
wrg
ααα α α
∞
+= +
− = −+∑
1
11
( ) 1( )'( )
dndm
n dn dm dn
wrg
ααα α α
∞
−= −
−=
− +∑
It yields the system in the unknowns and m ma b :
2 2
1
( ))( ) ( )1'( ) '( )
pdm dmm n o
ndm dm dn dm dm p
LL Lb a Fg g
αα αα α α α α α
∞−+ +
=
− −+ = −
− + − +∑
2
1
( ) 1 0'( )
dmn m
ndm dm dn
L b ag
αα α α
∞−
=+
− + =+∑
where
( ) ( ) ( )L f gα α α± ± ±=
For the presence of the kernel 1
dm dnα α+, the numerical solution of the previous system
does not introduce any difficulty and the sum can be approximate with few terms. Alternatively we can reduce the order of the system by eliminating the unknowns mb or
ma but it does not produce significant advantages .
The samples 2 ( )'( )
dmm
dm
wag
αα+
= and 1( )'( )
dmm
dm
wbg
αα−
−=
− provide the two meromorphic
functions 1 ( )r α− and 2 ( )r α+ . 1.5.5.2 Some Classes of matrices with Explicit Factorizations
9
1.5.5.2.1 Matrices having rational eigenvectors
All the matrices may be represented by the Jordan form: (77) 1)()()()( −= αααα SJST where )(αS is the eigenvectors matrix and )(αJ is a Jordan matrix having the form [40]: (78) ),....}(),({)( 2211
λλα mm JJquasidiagJ = In eq.(78) )(
1 imJ λ is the ii mm × Jordan block relevant to the eigenvalue )(αλλ ii = (with multiplicity im ) of )(αT :
(79)
i
i
i
i
imiJ
λ
λλ
λ
λ
..0001............000..100..01
)( =
The factorization of )(αT can be accomplished explicitly (in a weak form) if the matrix
)(αS is rational. In fact, in this case, eq.(77) reduces the factorization of )(αT to that of the Jordan matrix )(αJ , that, in general, is triangular and can be accomplished with the method reported in section 1.4.2 Example 1: Factorization of the matrix M(α) considered in [50]
(80) )()()]()([
)]()([)()(1)( 22
22
22 αηαδαααδηαααδηααδηαα
ηαα
dddd
M+−
−++
=
where )(αd and )(αδ are functions of α and the parameter η is constant. It is possible to obtain:
(81) 11
)( αη
ηα
α −=S ,
)(00)(αδ
αdJ = , 2
2
221 1)(
αηαηηα
ηαα
−+
=−S
Consequently, it is possible to achieve a weak factorization of eq.(80) by factorizing the diagonal matrix J.
10
Example 2: Matrices of order two having the canonical form 1)(
)(1)(
αα
αb
aG =
These can be (weakly) factorized in an explicit form if )()(
)()(
2
2
αα
αα
pn
ba
=
where )(αn and )(αp are polynomials. In this case )(αS is rational since we have:
(82) 11
)()(
)()(
11)()(
)()(
)( αα
αα
αα
αα
α pn
pn
ba
ba
S −=−=
Besides the general method indicated in 1.4.2 for triangular matrices, the factorization of the matrix )(αJ can be accomplished by the logarithmic decomposition of )]([ αTLog by taking into account that [40] a generic function of a matrix )(Tf is given by:
)()],....}([)],([{)()]([ 12211
αλλαα −= SJfJfquasidiagSTf mm where
)()],....}([)],([{)()]([ 12211
αλλαα −= SJfJfquasidiagSTf mm with
(83)
)(..0001............)(00
)()!2(
1..)('!1
1)(0
)()!1(
1..)("!2
1)('!1
1)(
)]([)2(
)1(
i
i
im
iii
im
iiii
im
f
f
fm
ff
fm
fff
Jfi
i
i
λ
λ
λλλ
λλλλ
λ−
−
−
−
=
Since it easy to ascertain that the decomposed matrices of )]([ imi
Jf λ :
+− += )]}([)]}([{)]([ imimim iiiJfJfJf λλλ
do commute, the following explicit factorization of )(αT follows:
)(],....})]}([exp[{],)]}([{exp[{)()]( 121 21
αλλαα −−−− = SJLogJLogquasidiagST mm
)(],....})]}([exp[{],)]}([{exp[{)()]( 1
21 21αλλαα −
+++ = SJLogJLogquasidiagST mm
11
1.5.5.3 Matrices Commuting with Rational Matrices Let us consider the matrices defined by: (84a) )()()()( 10 αααα RfRG += where )(0 αR and )(1 αR are rational in α and )(αf is an arbitrary function of α . The factorization of )()()( 000 ααα +−= RRR yields (84b) [ ] )()()()()(1)()( 0
101
100 ααααααα +
−+
−−− += RRRRfRG
By taking into account that )()()()()( 1
011
0 ααααα
dPRRR =−
+−
−
where )(αP and )(αd are respectively matrix and scalar polynomial functions, it is possible to reduce the factorization of the matrix )(αG to that of the matrix
)()()(1)( α
ααα P
dfW += . This last matrix commutes with the polynomial matrix )(αP
and can be explicitly factorized by using the procedure reported in section 1.4.4.5. According to Cayley’s theorem we have: (85) )()(...)()(1)()](log[ 1
11 ααψααψαψα −−+++= n
no PPW where n is the order of the matrix )(αW . If )(αP has distinct eigenvalues )(αλ i , the functions )(αψ i are obtained through the Sylvester formula[40, p.156]:
∑ ∏= ≠ −
−=
n
i
n
ij ji
ji
PgPg
1
1)(][)]([
λλλα
λα
that provides the formula:
(86) ∑ ∏= ≠ −
−+==+=
n
i
n
ij ji
ji
PdfPgP
dfW
1
1)(]
)()(1log[)]([])(
)()(1log[)](log[
λλλα
λαααα
ααα
The decomposition of (87) )()()( αψαψαψ +− += iii yields the factorized matrices (88) )(~)(~)( ααα +− ⋅= WWW
12
where: )]()(...)()(1)(exp[)](~ 1
)1(1 ααψααψαψα −−−−−− +++= n
no PPW (89) )]()(...)()(1)(exp[)](~ 1
)1(1 ααψααψαψα −+−+++ +++= n
no PPW This factorization process is effective since the factorized matrices
)(~)(~ αα +− WandW are both functions of the same matrix )(αP , and so they certainly commute. Furthermore, these factorized matrices are regular together with their inverses in the half-planes 0]Im[ ≤α and 0]Im[ ≥α respectively, since )(αP is polynomial in α and hence regular on the whole complex plane−α . When the polynomial matrix )(αP is constant in. ( ) oP Pα = , the factorized matrices have algebraic behavior ∞→α . We obtain again the results of 1.5.5.2.1. Conversely, if the polynomial matrix )(αP isnot constant, the previous factorized formulas ensure only that the factorized matrices and their inverses are regular in the upper and lower −α half- planes, while they do not ensure that their asymptotic behavior in these half-planes is algebraic as ∞→α . This non-algebraic behavior introduces in the Wiener-Hopf procedure a transcendental, and not polynomial, entire vector function 1.5.5.3.1 Explicit expression of ( )iψ α According to Sylvester representation, in the case of simple eigenvalues )(αλλ ii = of
)(αP , the matrix log[ ( )] log[ ( )1 ( )]W h Pα α α= − may be so represented:
(96) ∑ ∏= ≠ −
−=
n
i
n
ij ji
ji
PWW
1
1)()](log[)](log[
λλλα
λα
Dealing with matrix of order n, it yields: (97) )()(...)()(1)()](log[ 1
11 ααψααψαψα −−+++= n
no PPW For instance, for a matrix )(αP of order two, it yields: (98) 1log[ ( ) ( )] ( )1 ( ) ( )oh P Pα α ψ α ψ α α− = +1 where:
1 2 2 1
1 2 2 1
log[ ( ) ] log[ ( ) ]( )oh hα λ λ α λ λψ αλ λ λ λ
− −= − −
− −,
13
1 21
1 2 2 1
log[ ( ) ] log[ ( ) ]( ) h hα λ α λψ αλ λ λ λ
− −= +
− −
For matrices of order three: (99) 2
1 2log[ ( ) ( )] ( )1 ( ) ( ) ( ) ( )oh P P Pα α ψ α ψ α α ψ α α− = + +
1 2 3 2 1 3 3 1 2
1 2 1 3 2 1 2 3 3 1 3 2
log[ ( ) ] log[ ( ) ] log[ ( ) ]( )( )( ) ( )( ) ( )( )o
h h hα λ λ λ α λ λ λ α λ λ λψ αλ λ λ λ λ λ λ λ λ λ λ λ
− − −= + +
− − − − − −
1 2 3 2 1 3 3 1 2
11 2 1 3 2 1 2 3 3 1 3 2
log[ ( ) ]( ) log[1 ]( ) log[ ( ) ]( )( )( )( ) ( )( ) ( )( )h hα λ λ λ λ λ λ α λ λ λψ αλ λ λ λ λ λ λ λ λ λ λ λ
− + − + − += − − −
− − − − − −
31 22
1 2 1 3 2 1 2 3 3 1 3 2
log[ ( ) ]log[ ( ) ] log[ ( ) ]( )( )( ) ( )( ) ( )( )
hh h α λα λ α λψ αλ λ λ λ λ λ λ λ λ λ λ λ
−− −= + +
− − − − − −
For matrices of order four: (100) 2 3
1 2 3log[ ( )1 ( )] ( )1 ( ) ( ) ( ) ( ) ( ) ( )oh P P P Pα α ψ α ψ α α ψ α α ψ α α− = + + + where:
1 2 3 4 2 1 3 4
1 2 1 3 1 4 2 1 2 3 2 4
3 1 2 4 4 1 2 3
3 1 3 2 3 4 4 1 4 2 4 3
log[ ( ) ] log[ ( ) ]( )( )( )( ) ( )( )( )
log[ ( ) ] log[ ( ) ]( )( )( ) ( )( )( )
oh h
h h
α λ λ λ λ α λ λ λ λψ αλ λ λ λ λ λ λ λ λ λ λ λ
α λ λ λ λ α λ λ λ λλ λ λ λ λ λ λ λ λ λ λ λ
− −= − −
− − − − − −
− −− −
− − − − − −
1 2 3 2 4 3 4 2 1 3 1 4 3 4
11 2 1 3 1 4 2 1 2 3 2 4
3 1 2 1 4 2 4 4 1 2 1 3 2 3
3 1 3 2 3 4 4 1 4 2
log[ ( ) ]( ) log[ ( ) ]( )( )( )( )( ) ( )( )( )
log[ ( ) ]( ) log[ ( ) ]( )( )( )( ) ( )( )(
h h
h h
α λ λ λ λ λ λ λ α λ λ λ λ λ λ λψ αλ λ λ λ λ λ λ λ λ λ λ λ
α λ λ λ λ λ λ λ α λ λ λ λ λ λ λλ λ λ λ λ λ λ λ λ λ
− + + − + += + +
− − − − − −− + + − + +
+ +− − − − − 4 3 )λ λ−
1 2 3 4 2 1 3 4
21 2 1 3 1 4 2 1 2 3 2 4
3 1 2 4 4 1 2 3
3 1 3 2 3 4 4 1 4 2 4 3
log[ ( ) ]( ) log[ ( ) ]( )( )( )( )( ) ( )( )( )
log[ ( ) ]( ) log[ ( ) ]( )( )( )( ) ( )( )( )
h h
h h
α λ λ λ λ α λ λ λ λψ αλ λ λ λ λ λ λ λ λ λ λ λ
α λ λ λ λ α λ λ λ λλ λ λ λ λ λ λ λ λ λ λ λ
− + + − + += − −
− − − − − −
− + + − + +− −
− − − − − −
1 2
31 2 1 3 1 4 2 1 2 3 2 4
3 4
3 1 3 2 3 4 4 1 4 2 4 3
log[ ( ) ] log[ ( ) ]( )( )( )( ) ( )( )( )
log[ ( ) ] log[ ( ) ]( )( )( ) ( )( )( )
h h
h h
α λ α λψ αλ λ λ λ λ λ λ λ λ λ λ λ
α λ α λλ λ λ λ λ λ λ λ λ λ λ λ
− −= +
− − − − − −
− −+ +
− − − − − −
14
The complexity of these expressions is due to the fact that even though we have )(αP rational, the eigenvalues )(αλ i are not rational functions of α . If the order of the matrix )(αP does not exceed 4, MATHEMATICA provides analytical expressions of
)(αλ i . Alternative we can use the following technique to obtain the functions ( )iψ α . This technique avoids the introduction of the eigenvalues of ( )P α . Since
1( ) ( )( ) 1 ( ) ( ) 1 ( )( ) ( )
f fW P P Pd d
α αα α α αα α
− = + = − − =
it is not restrictive to refer to the factorization of the matrix :
1( ) 1 ( ) ( )G l Pα α α−= − where ( )l α is a scalar known function: We have:
( )1
0
1[ ( ) ( ) 1 ]1 ( )
lLog l P dx
x Pα
α αα
−− + =−∫
Give a matrix P of order n, we will evaluate the matrix 1( )x −−1 P in terms of rational expressions of x. For illustrative purposes we will consider only the case n=4. By Cayley theorem we have: (121) 1 2 3
1 2 3( ) ( ) ( ) ( ) ( )ox r x r x r x r x−− = + + +1 P 1 P P P The rationality of the four functions ( )ir x ( 0,1,2,3i = ) is evident an in the following we will evaluate them explicitly. We have: (122) 2 3
1 2 3( ) ( ) ( ) ( ) ( )ox r x r x r x r x − + + + = 1 P 1 P P P 1 Taking into account the characteristic equation: (123) 4 3 2
1 2 3 4 0a a a a+ + + + =P P P P 1 , eq. (122) yields:
15
(124) 2 3
1 2 3( ) ( ) ( ) ( ) 0oe x e x e x e x+ + + − =1 P P P 1 where:
0 4 3 0 1 3 3 0 1
2 1 2 3 2 3 2 1 3 3
, ,,
e a r r x e a r r r xe r a r r x e r a r r x
= + = − += − + + = − + +
To evaluate 1 2 3, , ,oe e e e , equation (124) implies the four homogeneous equations in the unknowns 1 2 31, , ,oe e e e−
2 31 2 3( ) 1 ( ) ( ) ( ) 0 ( 1,.., 4)o i i ie x e x e x e x iλ λ λ− + + + = =
where ( 1,2,3,4)i iλ = are the eigenvalues of P , Since the determinant of coefficients (Vandermonde determinant) is not vanishing it yields: (126) 1 2 31 0, 0, 0, 0oe e e e− = = = = Consequently (125) provides the evaluation of 1 2 3( ), ( ), ( ), ( )or x r x r x r x It yields the rational expression:
3 2 21 2 3 1 2
0 14 3 2 4 3 21 2 3 4 1 2 3 4
12 34 3 2 4 3 2
1 2 3 4 1 2 3 4
( ) , ( )
1( ) , ( )
x a x a x a x a x ar x r xx a x a x a x a x a x a x a x a
x ar x r xx a x a x a x a x a x a x a x a
+ + + + += =
+ + + + + + + +
+= =
+ + + + + + + +
where the coefficient , 1, 2,3,4ia i = are the coefficients of the characteristic equation
4 3 21 2 3 4 0a a a a+ + + + =P P P P 1
The Bocher equations provides the evaluations of the coefficients ia of the characteristic equation in terms of the entries of the matrix P . In fact we have:
2 1 1 2 31 1 21 1 2 3
43 1 2 2 1 3 44
, , ,2 3
( 1) det[ ]4
a s a s sa s sa s a a
a s a s a s sa P
+ ++= − = − = −
+ + += − = −
where:
16
[ ] 21 2
3 43 4
, ,
,
s Tr P s Tr P
s Tr P s Tr P
= = = =
It shows that for polynomials matrices P , the coefficients ia are polynomials of α and the functions
1 1
2 2 3 3
( ) ( , ), ( ) ( , ),( ) ( , ), ( ) ( , )
o or x r x r x r xr x r x r x r x
α αα α
= == =
are rational function of the algebraic functions
4 3 21 2 3 4( , ) ( ) ( ) ( ) ( )f x x a x a x a x aα α α α α= + + + +
Of course the value of x that constitute poles of the functions
1 2 3( ), ( ), ( ), ( )or x r x r x r x are the zeroes with respect x of the algebraic equation ( , )f x α and represent the eigenvalues of P .
From the equation ( )
0
1[ ( )]1 ( )
lLog G dx
x Pα
αα
=−∫
It follows:
( ) ( ) ( )1 21 20 0 0
( ) 330
2 30 1 2 3
log[ ( ) ( ) 1 ] ( , ) ( , ) ( ) ( , ) ( )
( , ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
l l l
o
l
l P r x dx r x dx r x dx
r x dx
g g g g
α α α
α
α α α α α α α
α α
α α α α α α α
−− + = + + +
+ =
= + + +
∫ ∫ ∫∫
1 P P
P
1 P P P
where:
( )
0( ) ( , ) 0,1, 2,3
l
i ig r x dx iα
α α= =∫
Asymptotic behavior of the logarithmic representation of 1( ) ( ) 1l Pα α−− + We have obtained the equation:
1
2 31 2 3
( ) ( ) 1[ ( )1 ( ) ( ) ( ) ( ) ( ) ( )]o
l PExp g g g gα α
α α α α α α α
−− + =
= + + +P P P
17
Taking into account that the first member is a algebraic as α → ∞ : 1( ) ( ) 1f P Mνα α α−− + ≈
where M is a constant matrix, it follows:
[ ( )1]oExp g να α→ , or ( ) logog α ν α→
1 1[ ( ) ( )] [ ( ) ]pExp g P Exp g να α α α α→ → , or 1log( ) pg ν ααα
→
where p is the degree of ( )P α and so on . It will provide the proof of the convergence of the integrals that will introduced later.
A Procedure to Eliminate the Exponential Behavior Let us consider, for sake of simplicity, that the degree of )(αP does not exceed 2, remembering that this case is very important since e.g. it is related to the problem of diffraction by an half plane having arbitrary impedance faces immersed in an isotropic medium. The logarithmic decomposition yields: { }{ } { } { } { }
1
2 30 1 2 3
log[ ( ) ( ) 1 ]
( ) ( ) ( ) ( ) ( ) ( ) ( )
l P
g g g g
α α
α α α α α α α
−
±
± ± ± ±
− + =
+ + +1 P P P
In order to achieve the behavior of the functions { }( ) , 0,1, 2,3ig iα
±= for ∞→α , it is
important to introduce the following identity:
(106) ( )αααααα −
+−−−=− −
−
−
−
uuuu
u N
N
N
N
1
1
1
2
2
11
in the decomposition formulae (sect.1.3):
{ }1,2
( )1( ) 0,1, 2,32
ii
g ug du ij uγ
απ α±
= ± =−∫
It yields:
18
{ }10
0 2
1( )g Oψαα α±
= ± − +
{ }11
1 2
1( )g Oψαα α±
= ± − +
(107){ }1 2 32 2 2
2 2 3 4
1( )g Oψ ψ ψαα α α α±
= ± − − − +
{ }1 2 3 4 53 3 3 3 3
3 2 3 4 5 6
1( )g Oψ ψ ψ ψ ψαα α α α α α±
= ± − − − − − +
where:
1 10 0 0
1( ) ( )2
G g u duj
ψ ψπ
∞
−∞= = ∫
1 11 1 1
1( ) ( )2
G g u duj
ψ ψπ
∞
−∞= = ∫ ,
(108) 1 12 2 2
1( ) ( )2
G g u duj
ψ ψπ
∞
−∞= = ∫ , 2 2
2 2 21( ) ( )
2G u g u du
jψ ψ
π∞
−∞= = ∫ ,
3 3 22 2 2
1( ) ( )2
G u g u duj
ψ ψπ
∞
−∞= = ∫
1 13 3 3
1( ) ( )2
G g u duj
ψ ψπ
∞
−∞= = ∫ , 2 2
3 3 31( ) ( )
2G u g u du
jψ ψ
π∞
−∞= = ∫ ,
3 3 23 3 3
1( ) ( )2
G u g u duj
ψ ψπ
∞
−∞= = ∫ ,
4 4 33 3 3
1( ) ( )2
G u g u duj
ψ ψπ
∞
−∞= = ∫ , 5 5 4
3 3 31( ) ( )
2G u g u du
jψ ψ
π∞
−∞= = ∫
As we have: (102) 2
1 2( ) oP α α α= + +A A A (103) 2 2 3 4( )P α α α α α= + + + +o 1 2 3 4B B B B B (104) 3 2 3 4 5 6
5 6( )P α α α α α α α= + + + + + +o 1 2 3 4C C C C C C C
19
with , ,i i iA B C constant matrices of order four. In the expression: { }{ } { } { } { }
1
2 30 1 2 3
log[ ( ) ( ) 1 ]
( ) ( ) ( ) ( ) ( ) ( ) ( )
l P
g g g g
α α
α α α α α α α
−
±
± ± ± ±
− + =
+ + +1 P P P
The term { } g ( )o α
± does not involve offending exponential behaviors.
The term { }1( )g α
± involves the offending contribution
12 1
2ψαα
A∓ ,
the term { }2 ( )g α
± involves the offending contributions
1 1 2 1 2 3
2 3 42 2 2 2 2 22 3 42 2 3( ) ( )ψ ψ ψ ψ ψ ψα α α
α α α α α α+ + +B B B∓ ∓ ∓ ,
and, finally, the term { }3 g ( )α
± involves the offending contributions
(111) 1 1 2 1 2 3
2 3 43 3 3 3 3 32 3 42 2 3
1 2 3 4 1 2 3 4 55 63 3 3 3 3 3 3 3 3
5 62 3 4 2 3 4 5
( ) ( )
( ) ( )
ψ ψ ψ ψ ψ ψα α αα α α α α αψ ψ ψ ψ ψ ψ ψ ψ ψα αα α α α α α α α α
+ + + +
+ + + + + + +
C C C
C C
∓ ∓ ∓
∓ ∓
These expressions show that there is not offending behavior if the dominant terms relevant to { }1,2,3 g ( )α
±were vanishing:
11 ( ) 0Gψ =
(112) 1 2 32 2 2( ) 0, ( ) 0 ( ) 0G G Gψ ψ ψ= = =
1 2 3 4 53 3 3 3 3( ) 0, ( ) 0, ( ) 0, ( ) 0, ( ) 0G G G G Gψ ψ ψ ψ ψ= = = = =
Considering that we can multiply the given matrix for an arbitrary rational matrix ( )R α suggests to reduce the factorization of ( )G α to that of ( ) ( ) ( )eG R Gα α α= where ( )R α is a possible rational matrices that enforce the equations (see section 1.2) :
1 1 11 1 1( ) ( ) ( ) 0eG R Gψ ψ ψ= + =
1 1 1 2 2 22 2 2 2 2 2
3 3 32 2 2
( ) ( ) ( ) 0, ( ) ( ) ( ) 0
( ) ( ) ( ) 0e e
e
G R G G R GG R G
ψ ψ ψ ψ ψ ψ
ψ ψ ψ
= + = = + =
= + =
20
1 1 1 2 2 23 3 3 3 3 33 3 3 4 4 43 3 3 3 3 35 5 53 3 3
( ) ( ) ( ) 0, ( ) ( ) ( ) 0,
( ) ( ) ( ) 0, ( ) ( ) ( ) 0,
( ) ( ) ( ) 0
e e
e e
e
G R G G R GG R G G R GG R G
ψ ψ ψ ψ ψ ψ
ψ ψ ψ ψ ψ ψ
ψ ψ ψ
= + = = + =
= + = = + =
= + =
In the following we assume:
91
1
( ) [ ( ) ] jnj
jxα α−
=
= − +∏R P 1
where the scalars jx and the integer jn are to be determined. . By putting:
1 11 1 10
1[ ( ) ] ( ) ( , )2
xx f x r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
1 12 2 20
1[ ( ) ] ( ) ( , )2
xx f x r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
2 12 3 20
1[ ( ) ] ( ) ( , )2
xx f x u r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
3 1 22 4 20
1[ ( ) ] ( ) ( , )2
xx f x u r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
1 13 5 30
1[ ( ) ] ( ) ( , )2
xx f x r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
2 13 6 30
1[ ( ) ] ( ) ( , )2
xx f x u r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
3 1 23 7 30
1[ ( ) ] ( ) ( , )2
xx f x u r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
4 1 33 8 30
1[ ( ) ] ( ) ( , )2
xx f x u r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
5 1 43 9 30
1[ ( ) ] ( ) ( , )2
xx f x u r x u du dx
jψ α
π∞−
−∞− + = = ∫ ∫P 1
and forcing the nine conditions (114) yields the following system of non linear equations ( 9)q = : (116)
21
These equations can be rewritten in the abstract form: (117) 0=F(x) where:
(118) 1 2 3 4( , , , ,... )qx x x x x=x ,
1 11
2 21
1
( )
( )
......................
( )
qj
jj
qj
jj
qj
q j qj
n f x c
n f x c
n f x c
=
=
=
+
+
=
+
∑
∑
∑
F(x)
and its solution can be obtained again through the iterative process based on the Newton-Raphson formulae: (119) ( 1) ( ) ( ) 1 ( )[ )] ( )j j j j+ −= −x x J(x F x where the Jacobian matrix is:
( )
1 1 1 2 1 2 3 1 3 1
1 2 1 2 2 2 3 2 3 2( )
1 3 1 2 3 2 3 3 3 3
1 5 1 2 5 2 3 5 3 5 4 5
'( ) '( ) '( ) ... '( )'( ) '( ) '( ) ... '( )'( ) '( ) '( ) ... '( )( )
... ... ... ... ...'( ) '( ) '( ) '( ) '( ) i
q q
q qi
q q
q q
n f x n f x n f x n f xn f x n f x n f x n f xn f x n f x n f x n f xJ
n f x n f x n f x f x n f x=
=
x x
x
Despite the complexity of the problem, it must be noted that the Jacobian matrix can be
evaluated in closed form by the residue theorem since it involves integrals ∫∞
∞−of
rational functions of u . In fact we remember that: the scalars ( , )jr x u are the rational functions:
11 1 1 2 1 2 1 1 1
11 2 1 2 2 2 2 2 2
51 1 2 2 3
( ) ( ) .... ( ) ( )
( ) ( ) .... ( ) ( )
..............................................( ) ( ) .... ( ) ( )
q q
q q
q q q q q q
n f x n f x n f x G c
n f x n f x n f x G c
n f x n f x n f x G c
ψ
ψ
ψ
+ + + = − = −
+ + + = − = −
+ + + = − = −
22
3 2 21 2 3 1 2
0 14 3 2 4 3 21 2 3 4 1 2 3 4
12 34 3 2 4 3 2
1 2 3 4 1 2 3 4
( ) , ( )
1( ) , ( )
x a x a x a x a x ar x r xx a x a x a x a x a x a x a x a
x ar x r xx a x a x a x a x a x a x a x a
+ + + + += =
+ + + + + + + +
+= =
+ + + + + + + +
with the coefficient ( ), 1, 2,3, 4i ia a u i= = constituted by polynomial in u that are the coefficients of the characteristic equation 4 3 2
1 2 3 4 0a a a a+ + + + =P P P P 1 The Bocher equations provides the evaluations of the coefficients ia of the characteristic equation in terms of the entries of the matrix ( )u=P P . In fact we have:
2 1 1 2 31 1 21 1 2 3
43 1 2 2 1 3 44
, , ,2 3
( 1) det[ ]4
a s a s sa s sa s a a
a s a s a s sa P
+ ++= − = − = −
+ + += − = −
where:
[ ] 21 2
3 43 4
, ,
,
s Tr s Tr
s Tr s Tr
= = = =
P P
P P
As a matter of fact it is convenient to assume large arbitrary integers in so that the solution x of the non linear system is small. However we cannot assume ( 0 ) 0=x as starting point in the Newton-Raphson formulae since (0) 0=x yields a Jacobian matrix (0)( ) (0)J J=x not invertible. To obtain a good starting point (0)x , we approximate the function ( )if x with the MacLaurin expansion
(2) ( )(1) 2(0) (0)( ) (0) .....
2! !
qqi i
i if ff x f x x x
q≈ + + +
and obtain the starting point ( 0 )x by solving the algebraic system:
23
( ) ( ) ( )
( ) ( ) ( )
1 0 0 0 0 0 011 1 1 2 2 1 1 2 2 1
1 0 0 0 0 0 022 1 1 2 2 1 1 2 2 2
1 0 0 01 1 2 2
(0)(0)( .. ) .... .. 0!(0)(0)( .. ) .... .. 0!
........(0)
(0)( .. ) ....
q q q q
q q q q
q q q q
q q q q
q q q
ff n x n x n x n x n x n x cq
ff n x n x n x n x n x n x cq
ff n x n x n x
+ + + + + + + + + =
+ + + + + + + + + =
+ + + + + ( ) ( ) ( )0 0 01 1 2 2 .. 0
!q q q
q q qn x n x n x cq
+ + + + =
It yields:
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
1 1 10 0 01 1 2 2 1
2 2 20 0 01 1 2 2 2
0 0 01 1 2 2
..
..
........
..
q q
q q
q q q
q q q
n x n x n x s
n x n x n x s
n x n x n x s
+ + + =
+ + + =
+ + + =
where the quantities is are known. Notice that also these quantities can be evaluated by
the residue theorem since it involves integrals ∫∞
∞−of rational functions of α .
Assuming 1 2 .. qn n n N= = = = the system can be rewritten:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1 1 10 0 01 2 1
2 2 20 0 01 2 2
0 0 01 2
..
..
........
..
q
q
q q q
q q
N x x x s
N x x x s
N x x x s
+ + + = + + + =
+ + + =
Taking into account the Viete formulas it is possible to show that this system is equivalent to find the zeroes of a polynomial of degree q. Consequently explicit expressions of (0)x are possible in the case 4q ≤ . Alternatively if the component of (0)x are distinct and ordered according their absolute value 0 0 0
1 2 .... qx x x> > > , we can use the approximation:
24
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
1 1 10 0 0 11 2
2 2 20 0 0 21 2 1
1 1 10 01 2
01
..
..
........
q
q
q q q
q q
sx x xN
sx x xN
sx x
Ns
xN
−
− − −
+ + + =
+ + + ≈
+ ≈
≈
that yields immediate expressions of the starting point (0)x for using the the Newton-Raphson formulae:
( 1) ( ) ( ) 1 ( )[ )] ( )j j j j+ −= −x x J(x F x Notice also that we can choose the integers 1 2, .., qn n n such that 0
1 1x < .
For instance from the first equation we obtain:
0 021 10 0
1 11.. qx xs s qx x
N N N N N−
< + + < +
It yields:
101 1
1s
xN q
< <− +
provided that 1 1N s q> + −
On the reduction of the order of the system Particular form of the polynomial matrix ( )P α allows to reduce the order of the system. For instance if in 2
1 2( ) oP α α α= + +A A A the matrix 2A is diagonal, it is possible with a suitable normalization of the matrix kernel to factorize to obtain 2 =A 1 . Consequently also the matrix 4B and 6C are matrices unity: 4 =B 1 , 6 =C 1 and the offending behaviors due to the not vanishing 1 3 5
1 2 3, and ψ ψ ψ can be simple eliminate by multiplying the improper factorized by the scalars :
1 3 51 2 3exp[ ( + )]α ψ ψ ψ± +
In this case the reduction of the number of the unknowns is three.
25
In other case it is possible that even though 2 0≠A the matrices 4B and or 6C are vanishing. In these case again the order of the system reduces. If a kernel matrix ( )G α commutes with a polynomial Matrix ( )P α that can be factorized in the form
1 2( ) ( ) ( ).. ( )lP P P Pα α α α= , it is possible to reduce the factorization problem of ( )G α to the problem
of factorizing a matrix commuting with the polynomial matrix 2 3 4 1( ) ( ) ( ).. ( ) ( )lP P P P Pα α α α α or
3 4 1 2( ) ( ).. ( ) ( ) ( )lP P P P Pα α α α α and so on. Factorizations of ( )P α are always possible. For instance
by using the Smith representation we have )()()()( αααα FDEP = . Of course this transformation of the polynomial matrix produces advantages only if the new involved polynomial matrices are degree fewer than of degree of ( )P α .
Moreover we can pre(or post) multiplying ( )P α with an arbitrary polynomial matrix 2 ( )P α and the
factorization of ( )G α is reduced to the factorization of a matrix that commute with the polynomial
2 ( ) ( )P Pα α or 2( ) ( )P Pα α . However the degree of the new polynomial matrix is greater that the
degree of ( )P α and consequently this idea is not effective for reducing the order of the not linear system. Also to resort to the representation of rational matrices in the form (sect.1.5.1.1)
DBACR +−= −− 11 )1()( αα , can be not effective since even though the polynomial matrix 1 Aα − is linear in α , its order is the product of the degree and the order of ( )P α . For instance if ( )P α is of order 4 and degree 2 we obtain a matrix 1 Aα − of order 4 2 8× = . It involves a Cayley
representation involving matrices 7 ( )P α that increase considerably the number of the unknowns in the not linear system. Finally also the introduction of triangular representation (see sect. 1.5.3)
121 11
111 11 12
122 21 11 12
1 0( )
( ) ( ) 1
( ) 0 1 ( ) ( )0 ( ) ( ) ( ) ( ) 0 1
PP P
P P PP P P P
αα α
α α αα α α α
−
−
−
= ⋅
⋅−
is again not effective to reduce the order of the not linear system . The not linear equations as Jacobi inversion problem. The integral in u present in the functions ( )if x can be accomplished by the residue theorem. It yields that the functions ( )if x are abelian integrals involving the algebraic functions 4 3 2
1 2 3 4( , ) ( ) ( ) ( )f x y x a y x a y x a y x a= + + + + . Let’s suppose that the genus p related to the algebraic function is p and the abelian integrals are of first kind. If qp n= . Since the function 1,2,.., ( )pf x are constituted by a sum of Abelian integrals defined on the same range of integration , they can be expressed as (Bliss p.98) :
1 11 1 12 2 1( ) ( ) ( ) ... ( )p pf x c w x c w x c w x= + + + ,
2 21 1 22 2 2( ) ( ) ( ) .... ( )p pf x c w x c w x c w x= + + + .................................
26
1 1 2 2( ) ( ) ( ) ..... ( )p p p pp pf x c w x c w x c w x= + + + where ( )iw x 1, 2,3,..,i p= are p independent integrals of first kind in the Riemann surface of genus p. The system (110) can be rewritten Where:
11 11 12 1 1
2 21 22 2 2
1 2
..
.... .. .. .. .. ..
..
p
p
p p p pp p
m c c c cm c c c c
m c c c c
−
=
The new not linear system constitutes a Jacobi inversion problem well known in the literature. The existence of the solution is proved by topological methods (see for example: G. Springer, Introduction to Riemann Surfaces, Chelsea Publishing Company, New York, 1957). Several algorithms have been ideated for allowing the construction of the solution of Jacobi inversion problem (see for example Zverovich, E. I. ,Boundary value problems in the theory of analytic functions in Holder classes on Riemann surfaces. Russian Math. Surveys, 26, (1971) pp.117–192). The conceptual importance of these algorithms is that they provide the solution x of the not linear system by the zeroes of a polynomial of degree equal to the genus p . It means that if 4p ≤ we obtain an exact solutions constituted by radical of the coefficients of this polynomial. However the evaluation of these coefficient is very cumbersome. In general it involves abelian functions or Theta Riemann function of very complicated argument. To get an idea of this exact method works in the following section we consider extensively the case of matrix polynomial )(αP of order two. Let’s suppose now that the number of the not linear equations qn differs with the genus
p of the algebraic equation 4 3 21 2 3 4( , ) ( ) ( ) ( ) ( )f x x a x a x a x aα α α α α= + + + + related
to the polynomial matrix )(αP . The case qn p< reduces to a Jacobi inversion if we introduce qp n− additional unknowns 1, ,..,i qx i n p+= and the rational matrix defined by
1
1
( ) [ ( ) ] jp
nj
jxα α−
=
= − +∏R P 1
1 1 2 1 2 1 1
2 1 2 2 2 2
1 2
( ) ( ) .... ( )
( ) ( ) .... ( )
..............................................( ) ( ) .... ( )
p
p
p p p p p
w x w f x w x mw x w x w x m
w x w x w x m
+ + + = −
+ + + = −
+ + + = −
27
The previous procedure yields the following system of qn equations in qp n> unknowns: This system involve only qn Abelian integrals of first order. As the independent Abelian integrals of first order are p , we can add to qp n− arbitrary equations having the form: where the quantities 1, ,...i qm i n p+= can be arbitrary chosen . Conversely the case qn p> apparently cannot reduce to a Jacobi inversion problem. 1.5.10.3 Weakly factorization of a matrix commuting with a polynomial matrix In the following we report another technique for eliminate the offending behavior at α → ∞ Let’s consider a matrix ( )G α commuting with a polynomial matrix ( )P α : (1) 1( ) ( )1 ( ) ( )oG g g Pα α α α= + Let’s consider a point pα in the lower half-plane and suppose that for a suitable integer r
the matrix ( )( )r
p
P αα α−
is bounded. If n is the order of the matrix ( )G α according to
Cayley theorem we have:
(2) 1
1 1( ) ( )log[ ( )] ( )1 ( ) ( )
( ) ( )
n
o nr rp p
P PG α αα ψ α ψ α ψ αα α α α
−
−
= + +
− −
1 1 2 1 2 1 1
2 1 2 2 2 2
1 2
( ) ( ) .... ( )
( ) ( ) .... ( )
..............................................( ) ( ) .... ( )
q q q q
p
p
n n n p n
w x w f x w x mw x w x w x m
w x w x w x m
+ + + = −
+ + + = −
+ + + = −
1 1 11 2
1 2
( ) ( ) .... ( )
..............................................( ) ( ) .... ( )
q q q qn n n p n
p p p p p
w x w x w x m
w x w x w x m
+ + ++ + + = −
+ + + = −
28
The additive decomposition of the scalars ( )iψ α , yields the following weakly factorization:
(3) 1
1 ( 1)( ) ( )( ) exp ( )1 ( ) ( )
( ) ( )
n
o nr rp p
P PG α αα ψ α ψ α ψ αα α α α
−
− − − − −
= + +
− −
!
(4) 1
1 ( 1)( ) ( )( ) exp ( )1 ( ) ( )
( ) ( )
n
o nr rp p
P PG α αα ψ α ψ α ψ αα α α α
−
+ + + − +
= + +
− −
!
Now the problem is to eliminate the offending essential singularity pα that is present in
both the factorized matrices ( ) and ( ) G Gα α− +! ! . Notice that ( ) and ( ) G Gα α− +
! ! and their inverses possess algebraic behavior as α → ∞ . To this end rewrite the W-H equation in the form:
1 1( ) ( ) ( ) ( ) ( ) o
o
RG F G F Gα α α α αα α
− −+ + − − −= +
−! ! !
or:
1 1 1 1
31 22 3
1
( ) ( ) ( ) ( ) ( ) ( ) ( )
....( ) ( ) ( )
o oo o
o o
ss
sp p p p
R RG F G G F G G
R RR R
α α α α α α αα α α α
α α α α α α α α
− − − −+ + − − − − −
∞
=
− = + − = − −
= + + + =− − − −∑
! ! ! ! !
where the infinite unknowns vectors iR are the Laurent coefficients in the characteristic part of the essential singularity pα . Considering the first and the third member we have:
1
1( ) ( ) ( )
( )s o
oss p o
R RG F Gα α αα α α α
∞−
+ + −=
= +− −∑! !
or:
29
1 1 1
1( ) ( ) ( ) ( )
( )s o
oss p o
R RF G G Gα α α αα α α α
∞− − −
+ + + −=
= +− −∑! ! !
Now we decompose the known vector 1 1( ) ( ) oo
o
RG Gα αα α
− −+ − −! ! as:
1 1( ) ( ) ( ) ( )oo
o
RG G C Nα α α αα α
− −+ − = +
−! !
where ( )C α is the known characteristic part:
31 22 3
1( ) ....
( ) ( ) ( )s
ssp p p p
C CC CC αα α α α α α α α
∞
=
= + + + =− − − −∑
and ( )N α is regular in the essential singularity pα . By indicating with
31 22 3
0....
( ) ( ) ( )s
o ssp p p p
G GG GGα α α α α α α α
∞
=
+ + + + =− − − −∑
the characteristic part of 1( )G α−
+! plus the term 1( )oG G−
+= ∞! and taking into account the Cauchy product of two series yields the following characteristic part of
1
1( )
( )s
ss p
RG αα α
∞−+
= −∑! :
1
1 1 1
1( ) vector regular in ( ) ( )
ss
s i i ps ss s ip p
RG G Rα α αα α α α
∞ ∞−+ −
= = =
= + = − −
∑ ∑ ∑!
It yields the system :
1, 1, 2,...,
s
s i i si
G R C s−=
= = ∞∑
in the unknowns iR . Notice that the system has triangular form that allow to have an explicit recurrence expression of iR .
1
1.5.11 Factorization of a 2x2 matrix
A matrix of order two can be rewritten in the following canonical form:
(121) 1)(
)(1)(
αα
αb
aG =
There are many ways to reduce a general matrix of order two to the canonical form: e.g.
it is possible to rephrase the general matrix )()()()(
)(2221
1211
αααα
αmmmm
M = by considering
that:
(122)
)(00)(
1)()(
)()()(
)(1
)(00)(
)()()()(
)(
22
11
1122
21
2211
12
22
11
2221
1211
αα
ααα
ααα
αα
αααα
α
+
+
+−
+−
−
−=
==
mm
mmm
mmm
mm
mmmm
M
where
)()()( 111111 ααα +−= mmm )()()( 222222 ααα +−= mmm The factorization of )(αM is so reduced to that of the central matrix, that has a canonical form, where:
)()()(
)(2211
12
ααα
α+−
=mm
ma )()(
)()(
1122
21
ααα
α+−
=mm
mb
There are many other ways to reduce to canonical forms matrices of order two, since it is possible to modify the structure of the original given matrix )(αM to the matrix
)()()()( αααα +−= WMWG where )(α−W and )(α+W are (with their inverses) arbitrary minus and plus functions or arbitrary rational functions. The matrices )(α−W and )(α+W could be also rational matrices. In this circumstance accomplishing the factorization of
( ) ( ) ( )M M Mα α α− += yields to a weak factorization of the given matrix ( )G α The matrices of order two with canonical form have some interesting properties, e.g. if the left factorization is known:
(123) ++
++
−−
−−==ξςηϑ
ρτσω
αα
α1)(
)(1)(
ba
G
2
the right factorization by using equation [32] follows
(124) −−
−−
++
++
++
++
−−
−− ===ωτσρ
ϑςηξ
ξςηϑ
ρτσω
αα
α1)(
)(1)(
ba
G
A canonical matrix of order two 1)(
)(1α
αb
a is called a Daniele matrix when the
function )()(
αα
ba is a rational matrix of α , i.e:
(125) )()(
)()(
αα
αα
pn
ba
=
where )(αn and )(αp are polynomials. The Daniele matrices can be factorized explicitly [26,55] by using the equations
(126) )(ˆ)(ˆ1)(
)(1)( αα
αα
α +−== GGb
aG
where:
(127a)
)()()(21cosh[)()()(
21sinh[
)()(
)()()(21sinh[
)()()()()(
21cosh[
)(
]0)(
)(0)(
21exp[)()(ˆ
αααααααα
αααααααα
α
αα
ααα
pntpntnp
pntpnpnt
g
pn
tgG
−−
−−
−
−−−
=
==
(127b)
)()()(21cosh[)()()(
21sinh[
)()(
)()()(21sinh[
)()()()()(
21cosh[
)(
]0)(
)(0)(
21exp[)()(ˆ
αααααααα
αααααααα
α
αα
ααα
pntpntnp
pntpnpnt
g
pn
tgG
++
++
+
+++
=
==
In the previous expressions )(α−g and )(α+g are the factorized scalars of )(αG determinant: (128) )()()]([)( αααα +−== ggGDetg
3
The functions )(α−t and )(α+t follow from the decomposition of the function )(αt defined by:
(104) )()()()(1)()(1
ln)()(
1)( αααααα
ααα +− +=
−
+= tt
baba
pnt
Many matrices of order two have the Daniele form, for instance Hurd [49] showed that all the matrices of order two commuting with polynomial matrices reduce to matrices having the Daniele form. To obtain this result, let us consider the matrix )(αW
commuting with an arbitrary matrix polynomial )()()()(
)(2
1
αααα
αlnml
P = :
(129) )()(1)( ααα PfW += where )(αf is an arbitrary scalar and )(),(),(),( 21 αααα nmll are arbitrary polynomials in α . It may be observed that )(αW can be rewritten in the alternative form:
(130) )()(
011)(
)(1)()(
01)()(
1
αααα
αααα
mlba
mlsW
−
=
where:
)]()()[(211)( 21 αααα llfs ++= , )]()([
21)( 21 ααα lll −=
)()()(
ααα
sfa = , )]()()([
)()()( 2 ααα
ααα nml
sfb +=
This form shows that the factorization of the matrix )(αW reduces to that of the central
matrix 1)(
)(1α
αb
a. Since
)]()()([1
)()(
2 ααααα
nmlba
+= is rational in α , this central
matrix has the Daniele form. Alternatively, simple algebraic manipulations yield the alternative form:
(131) )()()()(
)(1)()(αααα
αααln
mlfsW
−+=
Matrices having this form are called Khrapkov matrices, since their explicit factorization has been studied in the paper by Khrapkov [23].
4
Given a matrix of order two dcba
G =)(α , there is a simple criterion to ascertain
whether it commutes with a rational matrix by looking at the eigenvector matrix S so defined:
(132) 112
4)(2
4)( 22
cbcdada
cbcdada
S+−+−+−−−
=
By introducing the indicator:
(133) adc
bda
bcdaSSi
−−
+−−=
−= −
22
4)(1
1001
2
1
it follows that, if the given matrix is rational, rr
rr
dcba
, except for the common irrational
factor rrrr cbda 4)(
12 +−
, the indicator matrix is rational. Since two commuting
matrices have the same S, and consequently the same indicator i, it follows that for a
given matrix dcba
G =)(α , commuting with a rational matrix , except for a common
factor, the functions b, c and da − must be rational. For instance, when it is applied to a
matrix having the form acba
M =)(α , this criterion indicates that the matrix M does
commute with a rational matrix if the ratio cb / is rational. However we remember that it is possible to deeply modify the structure of the original matrix ( )G α ,by pre or post multiplying by rational matrices and in this way sometimes it is possible to obtain, by suitable algebraic manipulations, matrices commuting with rational matrices starting from matrices that do not have this property. As a particular case of the considerations done in the section (On the reduction of the order of the system), it is important to observe that if the polynomial matrix ( )P α has
11( )P α and 22 ( )P α polynomials of second degree and 12 ( )P α and 21( )P α polynomials of first degree or constants, it is possible to eliminate very quickly the offending behavior at α →∞ . In fact a suitable normalization of the plus unknowns allow to have an equal coefficients c in the term containing 2α in 11( )P α and 22 ( )P α . So that putting:
1( ) [ ( )1 ( ) ( )]oW exp Pα ϕ α ϕ α α= +
5
the logarithm factorized matrices have the offending behavior
11exp[ ( ) ( ) ]P dα ϕ α αα
∞
−∞± ∫
This behavior can be eliminated by multiplying the logarithm factorized matrices by
the exponential scalar 1exp[ ( ) ]c dα ϕ α α∞
−∞∫∓
Example 4 Let us consider again the factorization of the matrix considered in [50]:
(133) )()()]()([
)]()([)()(1)( 22
22
22 αηαδαααδηαααδηααδηαα
ηαα
dddd
M+−−+
+=
where )(αd and )(αδ are functions of α and the parameter η is constant. The previous indicator i so stands:
)(221
1001
22
22
221
ηαηαηαηα
ηα −−−−−
+== −SSi
This shows that )(αM commutes with a rational matrix rr
rr
dcba
R =)(α having the
same indicator i. In order to find )(αR , let us choose: ηα== rr cb , 22 ηα +−=− rr da . By introducing for example 0=rd , the rational (or polynomial)
matrix commuting with )(αM so follows:
(134) 0
)(22
ηαηαηα
α+−
=P
The representation of )(αM follows: (135) [ ])()(1)()( αααα PfcM += By comparison with (110) this yields:
22
22 )()()(ηα
αηαδαα++
=dc ,
)()()()()( 22 αηαδα
ααδαd
df+−
=
By using (106) the factorization of )(αM is reduced to a Daniele matrix one where:
6
(136) )]()()[2(
)]()([22222 αηαδαηα
ααδd
da+−+−
−−= ,
)]()()[2(2])][()([2222
222
αηαδαηαηαααδ
ddb
+−+−+−−
=
222 )(4ηα +
=ba
In this case ba is rational too, so it is easier to obtain factorized matrices according to
the results that will be reported in the next section 1.5.11.1. By taking into account that the factorization of a given matrix )(αM may be reduced to the factorization of the matrix )()()()( αααα +−= WMWG , where )(α−W and )(α+W are (with their inverses) arbitrary minus and plus functions or arbitrary rational functions, it is possible to deeply modify the structure of the original matrix )(αM , and in this way it is sometimes possible to obtain, by suitable algebraic manipulations, matrices commuting with rational matrices starting from matrices that do not have this property. The factorized matrices )(ˆ α−G and )(ˆ α+G , defined by the equations (102), commute and they are regular (as are their inverses) in the half-planes 0]Im[ <α and 0]Im[ >α respectively. However, when the degree of the function )()( αα pn is larger than 2, the behavior of these factorized matrices is not algebraic at infinity. The study of this case was reported in [16] . The key idea in order to remove this unacceptable behavior, as it is indicated in section 1.5.10.1.2, is to rewrite the factorization equation in the form: (137) )(ˆ)()()(ˆ)(ˆ)(ˆ)( 1 ααααααα +
−−+− == GRRGGGG
where )(αR is an arbitrary rational matrix commuting with
0)(
)(0α
αp
n.
For example, it is possible to choose:
(138)
++++=
0)()(0
1...)()( 221 α
αααα
pn
xxxR o
where all the coefficients ix (i=0,1,…) are arbitrary constants. It is possible to factorize
)(αR by the previous procedure yielding: (139) )(ˆ)(ˆ)( ααα +−= RRR where )(ˆ)(ˆ αα +− RandR have non-algebraic behavior as ∞→α . As it has been reported in [16], it is possible to choose ix such that )(ˆ)(ˆ αα −− RG and )(ˆ)(ˆ αα ++ RG have algebraic behavior. By writing
7
(140) )(ˆ)(ˆ)()(ˆ)(ˆ)(ˆ)(ˆ)( 1 αααααααα ++−
−−+− == GRRRGGGG This reduces the factorization of )(αG to that of the rational matrix
)()()(1 ααα +−− = ii RRR , and it yields the final equations:
(141a) )()(ˆ)(ˆ)( αααα −−−− = iRRGG (141b) )(ˆ)(ˆ)()(ˆ αααα ++++ = GRRG i The factorization thus obtained could be weak, since the factorized matrices could have offending poles, but it is not difficult to overcome this problem (sect.1.5.9). Conversely, the big difficulties of this procedure are related to the search for the unknowns ix
(i=0,1,…). In fact, even though the factorized matrices )(ˆ α−R and )(ˆ α+R can be evaluated in closed form, the equations that must be satisfied by the unknowns ix (i=0,1,…). are not linear. Some detailed examples are reported in [16],[52],[53]. If a matrix of order two is in the Daniele form in [9] a proof of the existence of the solutions of the system 0),...,,( 121 =+ −NRii xxxcc has been provided. Moreover in the next section it will be presented a general theory that for the Daniele form matrices reduces the problem of the offending behavior as α →∞ to a Jacobi inversion problem. However, sometimes it could be better to consider the original form
)()(1)( ααα PhW += where )(αP is an arbitrary polynomial matrix of order two. For instance, dealing with the particular case )(αP of degree two, only one unknown is present, and, in this case, it is possible to obtain an analytical solution as it follows. Let us introduce the rational matrix (151) ( ) 1 ( )R xPα α= + Taking into account the expressions reported in sect. 1.5.1.0.2, to cancel the offending exponential behaviors we force the unknown parameter x to satisfy the non linear equation:
1 21 2
1 2 2 1 1 2 2 1
1 1log[1 ] log[1 ]log[1 ] log[1 ] ( ) ( )( ) x xf x d d
λ λλ λ ξ η ξ ηα α
λ λ λ λ λ λ λ λ∞ ∞
−∞ −∞
+ + + + = + = + − − − −
∫ ∫
where 1λ and 2λ are the eigenvalues of ( )P α . It yields:
8
(152) 21 2 2 1
( ) 11 ( )
df x ddx x x
αλ λ λ λ
∞
−∞
= + + + ∫
The quantities 1 2λ λ+ and 1 2λ λ are polynomials since are the trace and the determinant of matrix ( )P α . Consequently the integrand function is rational in α and so the residue theorem applies. In some important case [Hurd-Luneburg] the 1 2λ λ+ and
1 2λ λ are of degree 2 and the residue theorem yields
(153) )(
2)(23
2
dcxbxaxxkj
dxxdf
+++= π
where dcba ,,, are constants. By taking into account that 0)0( =f , )(xf can be expressed in terms of elliptical integrals, and, by inverting them, It is possible to get an explicit expression of x in terms of Jacobian elliptical functions. In fact, by introducing
))()(( 32123 xxxxxxadcxbxax −−−=+++
it yields
(154) 231
321
1
13
1312 )()(
,)()(
[[)(
21)(xxxxxx
xxxxxArcSinEllipticF
xxxaxxf
−−
−−
−=
Consequently, the equation (155) nxf =)( has the analytical solution:
(156) 2
231
3213121331
2
231
321312131
]])()(
),((21[[
]])()(
),((21[[
xxxxxxxxxxnJacobiSNxxx
xxxxxxxxxxnJacobiSNxx
x
−−
−+−
−−
−=
The factorization formulae follow from (157) +++
−−
−−−++
−−− == WRRRRWWRRRWW ~~][][~~~~~~ 111
9
where the factorization of 1
21 ))(1()( −− += ηη xPR can be obtained by the rational
matrices factorization method. Elimination of the offending exponential behavior for matrix of order two In sect. 1.5.11 matrix of order two commuting with polynomial matrices have been reduced to the Daniele form:
1)()(1
αα
ba
where )]()()([
1)()(
2 ααααα
nmlba
+= is a ratio between polynomial ( ) 1n α = and the
polynomial 2( ) ( ) ( ) ( )p l m nα α α α= + of degree N. For instance, for polynomial matrices of degree two, in general ( )p α is of fourth degree. Factorized matrices given by eqs.(127) have offending exponential behavior for the presence of the asymptotic term:
1
2
112
1 1 1 ( ) ( )1 1{ ln( ) 1 ( ) 1 ( ) ( )
exp1 ( ) ( ) 1 ( ) ( )1 1ln .... ln }
( ) 1 ( ) ( ) ( ) 1 ( ) ( )
1cosh ( ) ... sinh ( )( )
h
h
o h oh
a t b tdt
p p t a t b t
a t b t a t b tt tdt dtp t a t b t p t a t b t
c c ccp pp
α α
α α
α αα α α αα
∞
−∞
−∞ ∞
−∞ −∞
−
++
− = + + + + + − −
+ + =
∫
∫ ∫
112
1 11 12 2
...
( ) sinh ( ) ... cosh ( ) ...
hh
o h o hh h
cc
c c c cc cp p p
α α
α α αα α α α α α
−
− −
+ + + + + +
where h is the smaller integer greater or equal to / 2 1N − and :
1 ( ) ( )ln
( ) 1 ( ) ( )
i
ia t b ttc dt
p t a t b t∞
−∞
+=
−∫ , 0,..., 1i h= −
Without lack of generality in the following we will suppose N even so that:
12Nh = −
10
The presence of ic ( 0,..., 1i h= − ) originates an exponential offending behavior. In order to eliminate it we introduce rational matrices having the form:
R =
11( )
1 1
x p
x
α
They yield the offending behavior:
1
2
112
111 1 ( )1 1{ ln 1( ) 1 ( ) 1
( )exp
1 11 1( ) ( )1 1ln .... ln }1 1( ) ( )1 1( ) ( )
( ) ( )( ) 1cosh ( ) ... sinh( )
h
h
o hh
x p tdt
p p tx p t
x p t x p tt tdt dtp t p t
x p t x p t
r x r xr xpp
α α
α α
αα α α α
∞
−∞
−∞ ∞
−∞ −∞
−
+ + − = + + + + + − −
+ + =
∫
∫ ∫
112
1 11 12 2
( ) ( )( )( ) ...
( ) ( ) ( ) ( )( ) ( )( ) sinh ( ) ... cosh ( ) ...
o hh
o h o hh h
r x r xr xp
r x r x r x r xr x r xp p p
αα α α
α α αα α α α α α
−
− −
+ + + + + +
where:
11( )
( ) ln 1( ) 1( )
i
ix p ttr x dt
p tx p t
∞
−∞
+=
−∫ 0,..., 1i h= −
we have:
2
( ) 2( )
iid r x t dtdx p t x
∞
−∞=
−∫
This integrand function is rational in t and so the residue theorem yields terms having the form:
11
12 ( )'( ( ))
iu xp u x
−
where ( )u x is a suitable root of the fourth order equation:
2( ) 0p t x− = By putting 2 ( )x p t= , we can ascertain that the algebraic function 2( , ) ( )f u x p u x= − is of genus h and a Riemann surface (SpringerLink Encyclopaedia of Mathematics
(http://eom.springer.de/) can be associated to it . The quantities 12 ( )
'( ( ))
iu xp u x
−
( 1, 2,.., )i h= are abelian differential of first kind (Baker, p. 128). It means that by introducing a basis of Abelian differential 1 2, ,...., hd d dω ω ω (http://eom.springer.de/) in the Riemann surface, whatever are the picked poles in the residue theorem, we yield:
1
1 1 2 22
2( ) ( ) ( ) ........., ( ) ( 1, 2,.., )( )
ix
i i i ih htr x dx dt A x A x A x i h
p t xω ω ω
−∞
∞ −∞= = + + =
−∫ ∫
To eliminate the offending behavior let’ introduce the rational matrix:
1
1( )1
ih
ti
i
xpR
xα
=
=∏
It yields to the Jacobi inverse problem described in the previous section. Alternatively to the Jacobi inversion, we will work in the u-plane. The substitution ( )x u x u→ = yields integrals:
1 1
1/ 2
1
2 ( ) 2 '( ) ( )'( ) '( ) 2 ( )
( 1, 2,.. )( )
i i
i
u x dx p u u xdu dup u du p u p u
u du i hp u
− −
−
= =
= =
∫ ∫
∫
that is an hyper-elliptic integral of first kind (SpringerLink Encyclopaedia of Mathematics (http://eom.springer.de/) . Moreover the previous h-integral constitutes the
12
simplest basis of Abelian differential (SpringerLink Encyclopaedia of Mathematics (http://eom.springer.de/) in the Riemann surface relevant to the algebraic function
2( , ) ( )f u x p u x= − having genus h Let’ suppose that 1 2( ), ( ),.... ( )my x y x y x ( as iy x→∞ →∞ ) are the poles picked in the residue theorem. It yields:
1 21 1 1( ) ( ) ( )
( ) ...( ) ( ) ( )
mi i iy x y x y x
iu u ur x du du dup u p u p u
− − −
∞ ∞ ∞= + + +∫ ∫ ∫
Let’s indicate with:
11( )
( )
iy
iuab y dup u
−−
∞= ∫
the inverse of the abelian function ( 1( ( ))iab ab y y− = ): It follows:
1( ( )) ( ( ) .... ( ))i i i mab r x ab y x y x= + + Taking into account the summation theorem for the abelian functions, the elimination of the offending behavior yields algebraic equations that involve 1 2, ,... hx x x Some simple cases (esatto6_.nb) We have a relatively simple case when the polynomial ( )p α is biquadratic:
4 2( ) ( )p a bα α α= − + + It involves a genus 1h = and there is an only coefficient
01 ( ) ( )1 ln
( ) 1 ( ) ( )a t b t
c dtp t a t b t
∞
−∞
+=
−∫
that produces offending behavior. To eliminate it the rational matrix R =
11( )
1 1
x p
x
α
13
yields a similar coefficient 0
11( )1( ) ln 1( ) 1( )
x p tr x dt
p tx p t
∞
−∞
+=
−∫ that for a suitable value of
x could compensate 0c :
0 ( ) 0or x c+ = To obtain an explicit expression of 0 ( )r x , let’ consider:
02 4 2 2
3 31 1 2 2
( ) 2 2( )
2 224 2 4 2
d r x dt dtdx p t x t a t b x
jy a y y a y
π
∞ ∞
−∞ −∞= = − =
− + + +
= + + +
∫ ∫
where the residue theorem has been applied and 1y and 2y are the two solutions of the equation 4 2 2 0t a t b x+ + + = having negative imaginary part:
2 2
1
2 2
2
4 4 ,2
4 42
a a b xy
a a b xy
− − − −=
− + − −=
Substituting in the last member that evaluates 0 ( )d r xdx
yields:
2 2 2 20
2 2
1 1
4( ) 4( )( ) 2 24( )
a a b x a a b xd r x jdx a b x
π
− − + − + − − − +
= − +
It is evident that in the plane x there are not present the branch points 21 42
x a b= ± −
arising from 2 24( )a b x− + . The only branch points presents in 0 ( )d r xdx
are jb±
(pag.130boella.nb, biquadratica.nb). Multiplying by
14
2 2 2 2
1 1
4( ) 4( )a a b x a a b x
+ − + − + − − − +
numerator and denominator of the
previous equation after an algebraic manipulation yields (the results is correct if we exclude values of x near the not present branch points ):
0
22 2
( ) 11( )
2( )
d r x jdx ab x
b x b x
= − + − + + +
using the substitution
2z b x= + and taking into account that 0 ( ) 0r ∞ = , the integral yields [MATHEMATICA]
Known 0 0( )r x r= , the inversion of the previous equation provides the values of the x that allows to the rational matrix R to eliminate the offending behavior as α →∞ :
with:
22
a bma b−
=+
1.5.1.1. The )()(
αα
ba Rational Function of α case
15
If )()(
αα
ba is a rational function of α , the following slight modification of the Daniele
method allows to obtain directly factorized matrices having always algebraic behavior as ∞→α , avoiding, in this way, the cumbersome procedure described in [9], even if the
matrices thus obtained are factorized matrices only in the weak sense. By introducing
(158) )()(
)()(
2
2
αα
αα
pn
ba
=
the following representation of )(αG stands
(159)
)]()()(21cosh[)]()()(
21sinh[
)()(
)]()()(21sinh[
)()()]()()(
21cosh[
)(
]0
)()(
)()(0
)()()(21exp[)(]
0)()(0
)(21exp[)()( 2
2
αααααααα
ααααα
αααα
αα
αα
ααααα
αααα
pntpntnp
pntpnpnt
g
np
pn
tpngp
ntgG
−
=
===
where )(αg is the determinant of the matrix kernel )(αG and
(160) )()(1)()(1
ln)()(
1)(αααα
ααα
baba
pnt
−
+=
The decomposition of +−
−
++
−
+=
−
+
)()(1)()(1
ln)()(1)()(1
ln)()(1)()(1
lnαααα
αααα
αααα
baba
baba
baba
yields the weak factorization of )(αG 1.5.2. Extension to Matrices of Order 4 The identity:
(161)
−+
−+
−+
−+
−=
abab
abab
ab
abab
ba
abab
abb
a
11log
21cosh
11log
21sinh
11log
21sinh
11log
21cosh
11
1
holds even if faa = , fbb = are order 2 matrices and ba , scalars.
16
It is possible to rewrite this relationship in the form:
(162)
−
+
−
+
−
+
−
+
−=
bafbaf
bafbaf
np
bafbaf
pn
bafbaf
fbab
a
11log
21cosh
11log
21sinh
11log
21sinh
11log
21cosh
11
1 2
In some particular cases this equation provides the explicit factorization of order four
matrices if pn
ba= with n and p polynomials. In fact, by supposing that the involved
matrices commute, the matrix decomposition formula yields: (163)
( ) ( )
⊗−
⊗−=⊗−= ++−− 00
]}21exp[1{
00
]}21exp[1{
00
]}21exp[1{
11 222
pn
tfbap
ntfba
pn
tfbab
a
where, according to Cayley theorem:
+−
+−
+−
+=+=−
+=
+=+=
=+=−=
ttfttbafbaf
npt
fgLog
ggfggfbag
o
o
o
1
1
12
111log1
1][
11
ψψψψ
If f is diagonal, a commutative algebra is involved and it is possible to handle the matrices as scalars, while, if f is polynomial, it is possible to obtain commuting decompositions by assuming: (164) )(1 1 ηPttt o ±±± += (165) )(1 1 αψψψ Po ±±± += It yields the factorization formula: (166)
17
+
++
++
−−
−−
−
= gtnptnp
np
tnppntnp
tnptnpnp
tnppntnp
gb
a
21cosh
21sinh
21sinh
21cosh
21cosh
21sinh
21sinh
21cosh
11
If the involved polynomial have a degree 2≥ , both the factorized matrices −g , +g and
=
−−
−−
−
tnptnpnp
tnppntnp
W
21cosh
21sinh
21sinh
21cosh
,
=
++
++
+
tnptnpnp
tnppntnp
W
21cosh
21sinh
21sinh
21cosh
present exponential behaviors at ∞ . It is possible to eliminate these offending behaviors separately, taking into account the presence of commutating matrices:
(167) ++++−−
−−−−= rgRWRrRWrgfb
fa ~~~~~~~~1
1 11
where the rational matrices )(αRR = of order four and )(αrr = of order two will be determined according to the following considerations. The rational matrix R of order four have to be assumed such that −−RW ~~ and ++RW ~~ are algebraic. Suitable forms of this matrix are:
(168) ]1)]()()()[(
)]()()()[(1)(
21
21
+
+=
αααααααα
αPrrp
PrrnR
where )(2,1 αr are rational scalar functions.
As ∞→α , we have: ][)(21 0αα Oatnp +=− , ][)(
21 0αα Oatnp +−=+
where
22
221 ....)( −
−+++= NNaaaa αααα
and N is the degree of n by the degree of )(αP .
18
These offending behaviors of ±W may be canceled according to the procedure based on eq.(117). The commuting factorization of R involves the function
221 ))()()((1 ααα Prrpnrd +−= , this matrix commutes with )(αP . We first factorize
)( αgrd and then introduce a rational matrix of order two )(αr such that the offending behaviors of [ ]±)( αgrd are eliminated. 1.5.3. Extension to Order 2n Matrices The matrices having the form:
(169) 1
1fb
fa
with a and b scalars and f order n matrices, can decouple in a system of lower order if f presents many zeroes. For instance, if f has the form:
0021 ff
f =
we can ascertain that we are dealing with an order n system having the same form
11fb
fa. The same thing happens if f presents a vanishing column
If pn
ba= , where n and p are polynomials, it is possible to use the procedure indicated in
1.5.12 for matrices of order four. Alternatively, it may be observed that the matrix
11fb
fa commutes with the matrix
(170) fp
nfp
fnQ ⊗==
00
00
.
If f commute with a polynomial matrix, the problem we are dealing with is the one already considered in section 1.4.4.5, even if, in this case, we are dealing with factorized matrices showing exponential behavior at ∞→α , that have to be reduced to algebraic behavior via a cumbersome procedure. It is more interesting and important to consider those cases allowing explicit factorization constituted by matrices f such that the decomposition +− += ii
ii ddfϕ , with iϕ an
19
arbitrary scalar, involves two commuting decomposed matrices for every value of the positive integer i. In fact, by starting from the logarithm of the kernel:
(171) +−−
− +=++++= ψψϕϕϕϕ 1212
221 .......1]
11
log[ nno QQQ
fbfa
where −ψ and +ψ do commute. It is possible to observe that the decomposition of
22
1001
fnpQ ⊗= , 33
00
fp
nnpQ ⊗= , ….
reduce to that of (172) +− += ii
ii ddfϕ 12,..2,1 −= ni
that involves commuting decomposed term +− ji ded :
+−+− = ijji dddd , ( 12,..2,1 −= ni , 12,..2,1 −= nj ). By this way, the decomposition equation:
(173) +−−
− +=++++= ψψϕϕϕϕ 1212
221 .......1]
11
log[ nno QQQ
fbfa
where
(174a) .......}{1001
}{0
01 2
21 +⊗+⊗+= −−−− fnpfp
no ϕϕϕψ
(174b) .......}{1001
}{0
01 2
21 +⊗+⊗+= +++− fnpfp
no ϕϕϕψ
involves −ψ and +ψ , that commute. The commuting decomposition +− += ii
ii ddfϕ ( 12,..2,1 −= ni ) holds in the following
cases: -f is diagonal or it is a triangular Jordan matrix
20
- f commutes with a constant matrix - the Jordan decomposition: 1.. −= SdSf involves polynomial eigenvectors matrix S . - f commutes with a polynomial matrix If np behaves as )(αO as ∞→α , the first three cases involve factorized matrices having algebraic behavior. Conversely, there are serious problems dealing with the last case since, generally, the factorized matrices present exponential behaviors.
137
1.6 Approximate matrix kernel factorization 1.6.1 Rational approximation of the kernel All the powerful methods developed in functional analysis, i.e. iterative methods, moment methods and so on, can be effectively used for solving W-H equations or for obtaining suitable factorized matrices. Furthermore, the factorization problem could be faced with a suitable approximate method consisting in introducing approximate factorizations of the W-H equations kernels. This method is based on the observation that it is possible to factorize matrices with rational elements, so, the introduction of a rational approximation of an arbitrary kernel always allows to accomplish the factorization. 1.6.1.1 Zero-order approximation The W-H equation:
(1) o
oRFFG
ααααα
−+= −+ )()()(
has the following solution
(2) o
oo
RGGF
ααααα
−= −
−−++ )()()( 11
W-H solutions present many features that can be used for obtaining approximate solutions. For instance, if oα does not belong to the null space of )(1
oG α−+ , the presence
in the denominator of the factor oαα − in the solution (2), indicates that the dominant spectrum is for oαα ≈ . It suggests to introduce the approximation )()( 11
oGG αα −+
−+ ≈
that yields:
(3) )()()()()( 111 ααα
ααα
ααα ao
oo
o
ooo F
RG
RGGF =
−=
−≈ −−
−−++
This approximation represents the physical optical solution (PO). Conversely, when oα belongs to the null space of )(1
oG α−+ , typically we are dealing with
closed structures involving only a discrete spectrum. In this case we are usually interested in the evaluation of the residues of )(α+F in a simple pole op of )(1 α−
+G . The PO solution (3) provides the result:
138
(4)
[ ] [ ] [ ]oo
ooop
oo
oopp p
RGpGG
pR
GGFooo α
ααα
ααα−
=−
= −−−
−−−
−++ )()()(Res)()(Res)(Res 1111
It must be noticed that again the presence in the denominator of the factor oop α− , in the solution (4), indicates that the dominant spectrum is for oop α≈ . It suggests to put
1)()( 1 ≈−−− oo GpG α that yields the approximate expression:
(5) [ ] [ ]oo
opp p
RGFoo α
αα−
≈ −+ )(Res)(Res 1
Summing up, in the zero-order approximation (or physical optical solution) the kernel is approximated by a constant. In order to improve this approximation it is possible to introduce rational approximants for the kernels. These yield rational kernel that, in general, can be factorized in closed form. It has been ascertained that this procedure can be very efficient in obtaining accurate solutions of the Wiener-Hopf equations [37,54]. Example 1 In order to ascertain the accuracy of the physical optical approximation, let us consider the W-H equation
(6) o
FFGαα
ααα−
+= −+1)()()(
with:
(7) 22
1)(α
α−
=k
G
This equation is the W-H equation for the half-plane problem in the case of H-polarization when the source is a plane wave with direction oo k ϕα cos−= . The W-H technique (sect.1.2.4) yields the exact solution:
(8)o
o kkF
αααα
α−
−+=+ )(
The physical optical solution is given by:
(9) o
akF
αααα
−−
=22
)(
139
The relative error )(
)()()(
ααα
α+
+ −=
FFF
e a on the real axis α could be high (fig. 6.1)
fig. 6.1: Relative error of the PO solution when k=1-0.01j.
Looking for the far field solution in the ϕ direction, the asymptotic inverse Fourier Transform must be performed. The saddle point integration method yields that the scattering pattern by the half-plane is proportional to ))cos(( ϕπ −−+ kF [8, pag.141]. Fig.6.2 reports and compares the exact scattering )cos( ϕkF+ with the approximate one
)cos( ϕkFa + . Note a diverging behavior near 3πϕ ±= .
3πϕ = is the direction of the
reflected field; 3πϕ −= is the direction dividing the shadow region from the light region.
In these directions there is a strong coupling between the saddle point and a pole; the correction taking into account this circumstance yields to not diverging fields in these transition directions.
fig. 6.2: Comparison between the PO solution and the exact scattering by an half-plane
Note that even though the relative error could be not small, the diffraction phenomenon is overall described accurately, it means that the physical optical solution constitutes a reliable tool.
140
1.6.1.2 Pade approximants A better approximation, with respect to the physical optical one, can be obtained by use of the Pade approximants [37] of the kernel. MATHEMATICA provides this kind of approximants very easily. For instance, the instruction: (10) }],,0,{),([)( nmGPadeGp ααα = expresses the scalar kernel )(αG in terms of a rational function )(αpG where the order of the polynomials that constitute the numerator and the denominator are respectively m and n. The constant 0 means that this Pade approximant starts from the point 0=α where the approximation error is vanishing. The Pade approximant )(αpG may be very easily factorized: (11) )()()( oppp GGG ααα +−= and it provides the following approximate solution of the W-H equation (1):
(12) o
ooppa
RGGF
ααααα
−= −
−−++ )()()( 11
Example 2 In order to ascertain the accuracy of this approximation let us consider the previous
example, where: 01.01 jk −= , 2/32 koo =⇒= απϕ
22
1)(α
α−
=k
G
Choosing m=20 and n=22, MATHEMATICA provides very quickly a Pade approximant
)(αpG presenting the error )(
)()()(
ααα
αG
GGe p−
= reported in fig.6.3.
141
fig.6.3: Relative error of the Pade approximant for 01.01 jk −= , 2/32 koo =⇒= απϕ
a) 11 ≤≤− α , b) 1010 ≤≤− α The error is very small in the visible spectrum 11 ≤≤− α , fig.6.3 a), but increases considerably out of this interval fig.6.3 b). This is due to the fact that, for ∞→α ,
2)( −→ααpG , whereas 1)( −→ααG . It is very interesting to locate the position of the poles of the Pade approximant )(αpG . These are reported in fig. 6.4.
fig.6.4: Poles of the Pade approximant Since the structural spectrum is constituted by the two branch points k± , these poles are spurious. However, fig.6.4 shows that the location of them may simulate the two branch lines actually present in the problem. Fig. 6.5. a) reports the relative error
)())(
)(α
ααα
+
++ −=
FFF
e pp
142
fig.6.5 a): Relative error )(
))()(
ααα
α+
++ −=
FFF
e pp
The scattering pattern from the half-plane is reported in the fig. 6.5 b). This figure shows that there is not a considerable difference between the exact and the Pade approximated solution.
fig.6.5 b): Scattering pattern
By reducing the Pade parameters m and n the relative error )(
))()(
ααα
α+
++ −=
FFF
e pp
increases. For instance, choosing m=8 and n=10 it is double than with the previous choice. However, the scattering pattern does not change dramatically. Sometimes the parameter k, related to the branch point, can be arbitrary chosen. For instance, in the considered example, the scattering pattern does not depend on k. In these cases, since the rational approximation is better when the spectrum is far from the real axis α , it is convenient to assume k with a strong imaginary part, e.g. jk −= 1 . Fig. 6.6
illustrates the reduction of the relative error )(
))()(
ααα
α+
++ −=
FFF
e pp
143
fig. 6.6: different values of k: comparison of the relative errors
1.6.1.3 An interpolation approximant method The Pade approximant is accurate only in α points near the starting point sα , e.g. in the previous examples 0=sα has been chosen. It is possible to improve the approximation accuracy on a larger α range by using sophisticated techniques, such the multiple point Pade approximation. However, according to the authors, it is much more effective to introduce an interpolation method. The main idea of this approach is to force that the function:
(13) )(]1[])0([][11
αααα GbbGinm
mi
ii
m
i
ii ∑∑
+
+==
+−+=
is vanishing in a set of iα points suitable chosen: (14) 0][ =ii α nmi += ,.....,2,1 The mn + previous equations yield a system of mn + order on the ib , nmi += ,.....,2,1 unknowns. The solution of the system provides the following rational approximation of the kernel )(αG :
(15) mnmmm
mm
i bbbbbbGGG
αααααα
αα+++ +++
+++=≈
...1...)0(
)()( 221
221
By this way it is possible to obtain a rational approximant standing on a large range of α . In general, the rational approximations provided by this technique show a better accuracy with respect to those obtained by the Pade approximant.
144
1.6.1.4 Presence of discrete infinite spectrum. In this case the major error in the rational approximation of the kernel arises from the singularities near the real axis α . In order to dramatically improve the approximation it is convenient to rationalize only the part of the kernel containing the spectrum far from the real axis. For instance, let us consider the factorization of the kernel previously considered in the example 4 of section 1.3.2.2:
(16) )](sin[)sin()sin()(
cbcbG
+=
ττττα
Introducing the numerical data as in that section
24.2,
23.1,
21.1,2,1 λλλ
λπλ =+===== cbacbk
for vanishing values of the ]Im[k , we have two real zeroes and two real poles, consequently it is convenient to rationalize the function:
(17) ))(())((
)](sin[)sin()sin(
))(())((
)()( 21
221
2
22
221
2
21
221
2
22
221
2
cb
aa
cb
aa
cbcbaGM
αααααααα
ττττ
αααααααα
α−−−−
+=
−−−−
=
that presents only imaginary spectrum : ,.....)4,3( =± iaiα , ,.....)4,3,2( =± ibiα and
,.....)4,3,2( =± iciα . The Pade approximant of )(αM : (18) }}22,20,0,{),([_)( ααα MPadeMa =
yields the error )(
)()()(
xMxMxMxe a
m−
= reported in the fig. 6.7.
fig. 6.7: Relative error in Pade approximant of )(αM
145
By factorizing the rational function )(αaM , it yields the approximate factorized function:
(19) ))((
))(()()(
21
11
aa
cbaa MaG
αααααααα
α−−−−
= ++
The plots reporting the absolute values and the argument of the exact )(α+G and of the approximate )(α+aG for a large rang of α are indistinguishable. The following figures report the relative errors defined by:
)()()(
)(α
ααα
+
++ −=
GGG
e aabs
)()()(
)(α
ααα
+
+++
−=
GGG
e a
The relative error on the absolute values of the exact and approximate factorized functions is very negligible (fig. 6.8); for instance, for 2=α the relative error is
111008.6)2( −=abse
The biggest relative error is for )(
)()()(
ααα
α+
+++
−=
GGG
e a (fig.6.9), and it arises from the
relative error on the phase. It can be ascertain in fig. 6.10 that illustrates the arguments of the exact )(α+G and approximate )(α+aG
fig.6.8: Relative error on the absolute values of the plus factorized function
146
fig.6.9: relative error on the plus factorized function
fig.6.10: comparison between the exact and the approximate phase of the factorized plus function
1.6.1.5 Presence of continuous and of discrete spectra Dealing with truncated wave-guides the kernels involve continuous and discrete spectra. In these cases our experience suggest that the best way to obtain rational approximants is based on the interpolation method previously considered in 1.6.1.3. In the following the approximate rational factorization of the kernel
(20) ]cos[
)(d
eGdj
ττα
τ
= , 22 ατ −= k ,
factorized in 1.3.2.3 (Example 5), will be considered The exact factorized functions are: (21)
147
∏
∏
=
∞
=+
−
+−
+Γ−
−−
++−Γ
≈
−
+−
+Γ−
−−
++−Γ
=
bN
n
n
A
n
n
A
BnA
ABekdk
qk
jdAB
A
BnA
ABekdk
qk
jdAB
AG
1
1
1
1
1)cos(
]logexp[1
1
1
1)cos(
]logexp[1)(
αα
α
α
αατπτα
αα
α
α
αατπτα
α
αγ
αγ
To compare the exact and the approximate factorizations let us assume:
21.1),101(2,1 2 λ
λπλ =−== − bjk .
Since )(αG is an even function, the following even function has to be forced to be vanishing in the suitably chosen points iα :
(22) )(]1[])0([][12
1
)(2
1
2 αααα GbbGin
ni
nii
n
i
ii ∑∑
+
+=
−
=
+−+=
By assuming 10=n , in order to obtain the 2n+1 unknown bi,, we forced 0][ =mi in
12,.....,2,1 += nm The 21 previous equations yield a system of order 21 in the ib , 21,.....,2,1=i unknowns. The ib solution of the previous system, accomplished with MATHEMATICA, provides the following rational approximation of the kernel )(αG :
(23) 2221
412
211
2010
42
21
...1...)0(
)()(αααααα
ααbbb
bbbGGG a ++++++
=≈
Fig.6.11 reports the relative error )(
)()()(
ααα
αG
GGe aG
−= in the 1010 <<− α range.
This error is negligible far from the two singularities )01.01(28.6 jk −= and 07.060.51 j−=α , located very near to the real axis (fig. 6.12).
148
fig.6.11: Interpolation approximant: relative error in the range 1010 <<− α .
fig.6.12: Interpolation approximant: relative error in the range 75.4 << α .
The poles of the rational approximate kernel )(αaG are reported in fig.6.13:
fig.6.13: Spurious poles in the interpolation approximant
149
This figure clearly shows that the spectrum of )(αaG is different with respect to the spectrum of )(αG . However, since we are interested in the regular regions of the factorized functions, this fact does not induce significant errors. The factorization of the rational kernel )()()( ααα +−= aaa GGG has been accomplished by MATHEMATICA. Note that )(α++ −= aa GM , where the sign – is in influent on the factorization process, approximates very well the exact )(α+G by assuming a truncated product with Nb=200. In fig. 6.14 the absolute values of )(α+G and +aM show exactly the same pattern.
fig. 6.14: Absolute values of the exact and approximate plus factorized function
Fig. 6.15 and fig. 6.16 report the relative errors )(
)()()(
ααα
α+
++ −=
GMG
e a and
)](arg[)](arg[)](arg[
)(α
ααα
+
++ −=
GMG
e aph
fig.6.15: Relative error on the plus factorized function
150
fig. 6.16: Relative error on the phase of the plus factorized function
1.6.1.6 Approximate factorization of matrix of order two. Sometime it is not necessary to rationalize all the elements of the matrix kernel, but it is sufficient to approximate some elements of it in order to obtain matrices that can be factorized in closed form, e.g. the triangular matrices, Daniele-Khrapkov matrices and so on [54]. In the following the approximate factorization of a general matrix of order two will be considered. Without loss of generality, it can be written in the form:
(24) 2
1
dhhd
G =
where constant2
1 →dd as ∞→α
It can be rewritten:
(25) +
+
−
−=2
1
2
1
2
1
00
11
00
dd
ba
dd
dhhd
where, after the factorization of the scalars −+= 2,12,12,1 ddd , (1.4.11):
(26) −+
=12 dd
ha , +−
=12 dd
hb
In general the matrix 1
1b
a has not the Daniele-Khrakpov form. However, using the
same factorization expressions [7] of the Daniele-Khrakpov matrices it yields:
(27) ++−− ⋅⋅⋅= gMMgb
aWW1
1
151
where 21
2
1dd
hg −= and the matrices +WM , −WM and their inverse are given by:
(28) ( ) ( )
( ) ( )++
++±+ ±
±=
ttf
tftMW coshsinh1
sinhcosh1
(29)( ) ( )
( ) ( )−−
−−±− ±
±=
ttf
tftMW coshsinh1
sinhcosh1
where ±t are the decomposed function of
21
21
1
1log
21
11log
21
ddhdd
h
ababt
−
+
=−+
=
and
(30) 1
2
2
2
1
2
1
2
1
2
2
1
1
2
dd
dd
dd
dd
dd
dd
baf
=
===
+
+
−
−
+
+
−
−
The function f , given by eq. (30), offends the factorized matrices 1±
+WM and 1±−WM since
it makes them not regular in the half-planes 0]Im[ ≥α and 0]Im[ ≤α respectively. To obtain the true factorized matrices 1±
+M and 1±−M it is necessary to introduce an
approximation of the function f through a rational approximation of 1
2
dd
(31) ∏∏== −
−++
≈N
n n
nN
n n
no P
Zpz
cdd
111
2
)()(
)()(
)()(
αα
αα
αα
where all the zeroes nn Zz , and the poles nn Pp , are located in the half-plane 0]Im[ ≤α . It follows (32) =≈ ±
+±+
11WW MM
( ) ( )
( ) ( ))(cosh)(sinh)()(
)()(
)()(
)(sinh)()(
)()(
)()()(cosh
111
2
112
1
αααα
αα
αα
ααα
αα
ααα
++==+
+
+==+
++
∏∏
∏∏
−−
++
±
−−
++
±⋅
ttZP
zp
dcd
tPZ
pzc
ddt
N
n n
nN
n n
n
o
N
n n
nN
n n
no
152
(33)
( ) ( )
( ) ( ))(cosh)(sinh)()(
)()(
)()(
)(sinh)()(
)()(1
)()(
)(cosh
112
1
111
2
11
αααα
αα
αα
ααα
αα
αα
α
−−==−
−
−==−
−−
±−
±−
∏∏
∏∏
−−
++
±
−−
++
±
=≈
ttPZ
pz
dd
c
tZP
zp
cdd
t
MM
N
n n
nN
n n
no
N
n n
nN
n n
n
o
WW
Even 1±
+WM and 1±−WM are not true factorized function due to the presence of the poles
nn Zz , and nn Pp , , but the number of these points is finite so that these matrices are weak factorized matrices [7,14] and it is possible to eliminate these offending poles (sect.1.5.9) In order to do that let us introduce the homogeneous W-H system: (34) −
−−++ = iWiW mMmM 1
where the −im and +im vectors are true minus and plus functions with algebraic behavior at infinity. Dealing with well posed W-H systems of order two, there are two independent vectors −im and +im )2,1( =i . The evaluation of −im and +im )2,1( =i is important since these vectors provides true minus and plus factorized matrices 1±
+M and 1±−M through the equations (sect.1.2.9):
(35) [ ]++
−+ = 21
1 ,mmM [ ]−−− = 21 ,mmM It is possible to observe that −M and +M are approximate factorized matrices of M since
−−−
−++ = MMMM WW
11 . It implies:
MMMMMMM WWWW =≈= +−+−+− , Furthermore
111)det()det(det()det( ) =×== +−+− WW MMMM It implies that
costant)det(
1)det( ==−
+ MM
so that not only the (35) but also their inverses are plus and minus functions respectively (36) −
−−++ = iWiW mMmM 1
153
In order to solve eq.36, it is possible to observe that ++ iW mM has offending poles in
np− and nz− with unknown residues respectively given by 01
nC and 10
nD and
−−− iW mM 1 has offending poles in nZ and nP with unknown residues respectively given by
01
nA and 10
nB . In order to eliminate these offending poles, it is useful to rewrite eq.(36)
in the following form: (37)
=+
−+
−−
−−
− ∑∑∑∑====
++ 10
01
10
01
1111
N
n n
nN
n n
nN
n n
nN
n n
niW z
Dp
CP
BZ
AmMαααα
10
01
10
01
1111
1 ∑∑∑∑====
−−− +
−+
−−
−−
−=N
n n
nN
n n
nN
n n
nN
n n
niW z
Dp
CP
BZ
AmMαααα
This equation follows from the fact that the first member is regular in 0]Im[ ≥α and the second member is regular in 0]Im[ ≤α . By taking into account the algebraic behavior of
−im and +im , it yields that the third member is a constant vector; so, by forcing 02 =l and 01 =l , it is possible to obtain two independent solutions of the homogeneous equation (36). Equation (37) yields the following representation for −im and +im ,:
(38)
10
)(01
)(
10
)(01
)(10
01
)()(
11
1121
∑∑
∑∑
=−
=−
=−
=−−−
++
++
−+
−+
+=
N
nW
n
nN
nW
n
n
N
nW
n
nN
nW
n
nWi
Mz
DM
pC
MP
BM
ZA
llMm
αα
αα
αα
αα
αα
(39)
10
)(01
)(
10
)(01
)(10
01
)()(
1
1
1
1
11
121
1
∑∑
∑∑
=
=+
=
=+
=−
=
=+
=++
++
++
−+
−+
+=
N
nW
n
nN
nW
n
n
N
nW
n
nN
nW
n
nWi
Mz
DM
pC
MP
BM
ZA
llMm
αα
αα
αα
αα
αα
In the second member of (38) there are present plus functions m
m
ZA−α
, m
m
PB−α
and, in
addition, the offending poles mZ and mP involved in −WM . The regularity of −im in the half plane 0]Im[ ≤α imposes the vanishing of the residues of the second member of (38) for mZ and mP . It yields the following equations:
154
011
2 =−
+++
+ ∑∑==
N
n nm
nmm
N
n nm
nmm PZ
BRA
zZD
RlR
(40) (m=1,2..,N)
011
1 =−
+++
+ ∑∑==
N
n nm
nmm
N
n nm
nmm ZP
ASB
pPC
SlS
where:
( ))(tanh)()(
)()(
)()(
1
'
11
2m
N
n nm
nmN
n nm
nm
mo
mm Zt
ZZPZ
zZpZ
ZdcZdR −
==−
− ∏∏ −−
++
=
( ))(tanh)()(
)()(
)()(
1
'
12
1m
N
n nm
nmN
n nm
nm
m
mom Pt
PPZP
pPzP
PdPdcS −
==−
− ∏∏ −−
++
=
and ∏ ' means the product where it is put equal to 1 the vanishing factor.
By applying similar considerations to the equation (39) it yields the equations:
011
2 =+
−++−
+ ∑∑==
N
n nm
nmm
N
n nm
nmm Pp
BTC
zpD
TlT
(41)
011
1 =+
−++−
+ ∑∑==
N
n nm
nmm
N
n nm
nmm Zz
AUD
pzC
UlU (m=1,2..,N)
where:
(42) ( ))(tanh)()(
)()(
)()(
11
'
2
1m
N
n nm
nmN
n nm
nmo
m
mm pt
PpZp
ppzpc
pdpdT −
−−−−
+−+−
−−
−= +==+
+ ∏∏
(43) ( ))(tanh)()(
)()(
)()(
11
'
1
2m
N
n nm
nmN
n nm
nm
mo
mm zt
ZzPz
zzpz
zdczdU −
−−−−
+−+−
−−
−= +==+
+ ∏∏
By imposing 02 =l , the solution of the system (40)-(41) provides the solution −1m and
+1m of (34) through the representations (38) and (39). Similarly, by imposing 01 =l , it provides the independent solutions −2m and +2m , and, by this way, eq.s (35) define the approximate factorized matrices of G. Note that the factorized matrices are based on the rational approximation
(44) ∏∏== −
−++
≈N
n n
nN
n n
no P
Zpz
cdd
111
2
)()(
)()(
)()(
αα
αα
αα
155
In order to accomplish this approximation, Pade approximants [54] have been used in the past, however, as it will showed in the following examples (Part 3), it could be more convenient to use other techniques. The factorized matrices obtained by Pade or by an interpolation method provide an adequate approximation on the real axis, where they accurately evaluate also their derivatives. It follows that, according to an analytical continuation, these expressions can be used also in points sufficiently far from the real axis. This phenomenon occurs even working in the w-plane (sect.1.2.12.2) since the analytical continuation can be used also in improper sheets of the plane−α . However, this approximation introduces spurious poles that are not present in the original spectrum of the problem and does not have any physical meaning. This means that it is not possible to use the expression of the approximate factorized matrix near the spurious poles or, in general, outside the regularity half planes. In presence of the structural poles, such as those relevant to the surface waves, the complete solution of the problem requires the evaluation of the residues in poles with physical meaning and located outside the regularity half planes of the factorized matrices. In order to overcome this problem it is necessary to take into account both the exact and the approximate factorization of the kernel (45) +−= GGG (46) +−≈ aa GGG Let us suppose the presence of a singular point −α , i.e. for −G . Since G is exactly known and these singular point is located in the regularity region of +G , there are no problems to evaluate the residue of −G . In fact we have: (47) )(][Res)(][Res][Res 11
−−+=−
−+==−
−−−≈= αα
αααααα aGGGGG
A similar procedure applies for evaluating the residues of +G .
1.5.2.3 Moment method A powerful way to solve integral equation and in particular the W-H equation is to use the moment method in the spectral domain. To this end let’s consider the equation (17 ter) of ch.1.2:
( ) ( ) ( ) ( , )oG X X m Tα α α α α+ −= − ⋅ where:
156
( ) ( ) ( )( , ) [ , , ]dG G G tm t If td tα αα αα α
−= ==
−
To solve the previous equation we use the representation (moment representation):
( ) ( )i ii
X Cα ψ α+ +=∑
where the constant iC are unknown and the set of the plus function (expansion functions) are known A projection with the set of plus test functions ( )jϕ α+ provides the equations on the unknowns:
ji i ji
G C N=∑
where:
( ) ( ) ( )
( ) ( , )
ji j i
j j o
G G d
N m T d
ϕ α α ψ α α
ϕ α α α α
∞
+ +−∞
∞
+−∞
= − ⋅ ⋅
= − − ⋅ ⋅
∫∫
the minus function ( )X α− does not appear since in the performing the projection with the test functions, ( )jϕ α+ the Parseval theorem provides the equations:
( ) ( ) 0j X dϕ α α α∞
+ −−∞− ⋅ =∫
137
1.6.1 Solution of the W-H equations by integral equation In section 1.1.5, the W-H equation
(48) ( ) ( ) ( )o
RG F Xα α αα α+ −= +−
has been reduced to the Fredholm equation:
(49)[ ]
o
Rdtt
tFGtGj
FGααα
απ
αα−
=−
−+ ∫
∞+
∞−
++
)()()(.2
1)()(
The previous equation applies in the case where 0]Im[ <oα . A slight modification of the previous procedure (see next section) yields the following Fredholm integral equations for the cases where 0]Im[ >oα and Im[ ] 0oα = .
11
( ) ( ) ( )1( ) . ( ) , Im[ ] 02 o o
o
G G u F u RF du Gj u
αα α α
π α α α
−∞ + −
+ −∞
− + = >− −∫
1
11 1
( ) ( ) ( )1 1 1( ) . ( ) ( ) , Im[ ] 02 2 2
oo o
o o
G G u F u F RF du G Gj u
αα α α α
π α α α α α
−∞ + − −
+ −∞
− + = + =− − −∫
1
Similar equations can be obtained for the minus function ( ) ( )o
RF Xα αα α− −= +−
Putting 1
1( ) ( )G Gα α−= , we obtain:
11 1( ) ( ) ( )1( ) . , Im[ ] 0
2 oo
G G u F u RF duj u
αα α
π α α α
−∞ −
− −∞
− − = <− −∫
1
138
[ ]1 1 11
( ) ( ) ( ) ( )1( ) ( ) . ,2
o
o
G t G F t G RG F dtj t
α αα απ α α α
+∞ −− −∞
−− =
− −∫ Im[ ] 0oα >
1.6.1.1 Source pole oα with positive imaginary part Previously, dealing with the W-H equation
(142) o
oFFFG
ααααα
−+= −+ )()()(
the source pole oα has been assumed with negative imaginary part, 0]Im[ <oα . Conversely, let us now
deduce the Fredholm integral equation for the case when 0]Im[ >oα . If oα is not a pole of )(αG it follows (143) oo FFsG
o=+ ααα )]([Re)(
Since the integrand function is regular as α=x , when 0]Im[ >oα it is possible to assume valid, for continuity, the following equation:
(144) [ ]
0]Im[,)()()(
.2
1)()( >−
=−
−+ ∫ +
+ oo
oFdx
xxFGxG
jFG α
αααα
παα
γ
where the γ line differs from the real axis as shown in fig 6.31.
fig.6.43: Deformation of the integration path in presence of poles with positive imaginary parts
It follows (145)
[ ] [ ] [ ]oxx
xFGxGsdxx
xFGxGj
dxx
xFGxGj α
γ αα
αα
παα
π =
+∞
∞−
++
−−
−−
−=
−−
∫∫)()()(Re)()()(.
21)()()(.
21
and (146)
139
[ ] [ ] [ ]oo
o
o
o
o
x
FGGGxFsGG
xxFGxG
s o
o
)()()()]([Re)()()()()(
Re 1 ααααα
αααα
αα α
α
−+
=
+
−−
=−
−=
−
−
By substituting it yields (147)
[ ]0]Im[,)()()()()(.
21)()( 1 >
−=
−−
+ =∞
∞−
++ ∫ o
o
oo
FGGdx
xxFGxG
jFG α
αααα
αα
παα
The same procedure shows that when 0]Im[ =oα the following integral equation stands: (148)
[ ]0]Im[,)()(
21
21)()()(.
21)()( 1 =
−+
−=
−−
+ =∞
∞−
++ ∫ o
o
oo
o
o FGG
Fdx
xxFGxG
jFG α
αααα
αααα
παα
1.6.1.2 Validity of (49) for a particular W-H equation Dealing with the half-plane problem we have:
22
1)(α
α−
=k
G .
The W-H technique yields the exact solution:
o
o kkF
αααα
α−
−+=+ )(
Now, let us show that this solution satisfies the previous integral equation
Fig.6.17: Deformation of the original integration path
140
Introducing the suitable deformation of the integration path reported in fig. 6.17, it yields:
[ ] [ ]
dttt
ktk
dttt
tkk
tk
dttt
tkk
tkdt
ttFGtGdt
ttFGtG
o
o
o
o
o
o
))((12
))((12
))((12
)()()()()()(.
33
121
22
22
ααα
ααα
ααα
αα
αα
λλ
λλλ
−−+
+−=
−−−+
−−=
=−−−+
−−=
−−
=−
−
∫∫
∫∫∫ +
+∞+
∞−
+
By substituting ukt −= it yields:
[ ]
)(22
)/)(/(/1
112
))((12)()()(.
13
o
o
o
o
o
o
o
kkjj
dukuku
kuk
dttt
ktk
dtt
tFGtG
ααα
απααπ
ααα
ααα
αα
λ
−+
+−
−
=++
+
−−=
−−+
+−=
−−
∫∫∫∞∞+
∞−
+
This proves the validity of the integral equation (49). However, the numerical double
check is not easy. First of all αα
−−
tGtG )()( is indeterminate as α=t , so, it is necessary to
evaluate the limit for α→t . It yields the value '( )G α For what concerns the kernel
22
1)(α
α−
=k
G , the evaluation of this limit yields:
2 2 3/ 2
( ) ( ) '( )( )
G t G Gt k
α ααα α
−≈ =
− − as α→t .
A further difficulty arises from the fact that singularities of the kernel, e.g. the branch points, can be located near the real axis., but even this problem may be overcome as previously shown. Finally, the mayor difficulties are due to the presence of an integration defined over an infinite support that requires a large number of interpolating points in the quadrature formulas making very slow the achievement of an accurate value of the integral. In the next examples it will show the difficult to obtain an efficient numerical solution of the original Fredholm integral equation (49). However we will refine this method by introducing the F-kernels and by deforming the integration line in the plane−α or a different complex planes where the kernel presents a more suitable behavior. By this way the Fredholm technique becomes very efficient and it provides a reliable tool for solving the W-H equations. 1.6.1.1bis Physical meaning of the integral in the Fredholm equation
141
In the following we will show that the integral [ ]( ) ( ) ( )1 .2
G u G F udu
j uα
π α∞ +
−∞
−−∫ present
in the Fredholm equation, does not involve the pole oα of the source. In fact putting:
( ) regular term at = oo
TF xx
α αα+ = +
−
yields: (1.1)
[ ]
[ ]
( ) ( )1 .2
( ) ( )1 1 ( ) ( ) ( ) ( ). . ( )2 ( )( ) 2 ( )
( ) ( ) ( ) ( )1 ( ) ( ). [ ]2 ( )
1 . . [ ( , ) (2 ( )
o
o o o
o o
o o o
o
G u G T duj u u
G u G G u G G u GTdu Tduj u u j u u
G u G G GG u G Tduj u u u
PV m u m uj
απ α α
α α απ α α π α α α α
α α ααπ α α α α α
απ α α
∞
−∞
∞ ∞
−∞ −∞
∞
−∞
−=
− −
− − −= = − =
− − − − −
− −−= − − =
− − − −
= −−
∫
∫ ∫
∫( ) ( ) 1, )] . .
2 ( )o
oo o
G GTdu PV Tduj u
α ααπ α α α
∞ ∞
−∞ −∞
−⋅ + ⋅ −
− −∫ ∫
where:
(1.2) ( ) ( )( , ) G u Gm uu
ααα
−=
−
and both the integral have to be interpreted as principal values (P.V.) in the singular point x = ±∞ Now we have:
(1.3)2 2 2 2 2 2
2 2
1. . . . . .
1 122
o o
o o o o
o oo o
u uPV du PV du PV du duu u u u
du j ju
α αα α α α
α α π πα α
∞ ∞ ∞ ∞
−∞ −∞ −∞ −∞
∞
−∞
+= = + =
− − − −
= = − = −−
∫ ∫ ∫ ∫
∫
It yields: (1.4)
[ ]( ) ( ) ( ) ( )1 1 . . [ ( , ) ( , )]2 2 ( ) 2( )
oo
o o o
G u G G GT du PV m u m u Tdu Tj u u j
α α αα απ α α π α α α α
∞ ∞
−∞ −∞
− −⋅ = − ⋅ + ⋅
− − − −∫ ∫ Clearly the last expression shows that the integral is regular at oα α=
142
Th physical interpetration depends on the particular application. For instance, in diffraction problems the pole oα denotes a geometrical optic contribution. The obtained result shows that the integral in the Fredholm equation always has the physical meaning of a diffracted field.
1.6.1.3 Numerical solution of the W-H equations (49) Using a simple quadrature formula, the equation
[ ]
o
Rdtt
tFGtGj
FGααα
απ
αα−
=−
−+ ∫
∞+
∞−
++
)()()(.2
1)()(
may be approximated by the equation:
(50) o
hA
hAi
RihFihmj
hFGαα
απ
αα−
=+ +−=
+ ∑ )(),(.2
)()(/
/
where:
( ) ( ) ( )( , ) [ , , ]dG G G tm t If td tα αα αα α
−= ==
−
By forcing:
(51) }],,,{,)(),(.2
)()([/
/hAARihFihm
jhFGTable
o
hA
hAi−
−==+ +
−=+ ∑ α
ααα
παα
it yields a system of 12 +hA equations in the 12 +
hA unknowns
hAiihF ±±±=+ ,...,2,1,0),(
The solution of this system leads to the following representation of the plus function:
(52)
−
+−=≈ +−=
−++ ∑
o
hA
hAia
RihFihmj
hGFFαα
απ
ααα )(),(.2
)()()(/
/
1
The accuracy of the numerical solution depends on the introduced parameters A and h. Small values of h do not improve the accuracy and the sum becomes difficult to be evaluated, so, it is better to increase A rather than diminish h.
143
In order to ascertain the accuracy of this approximation, let us consider the previous
example where: 01.01 jk −= , 2/32 koo =⇒= απϕ
22
1)(α
α−
=k
G
Fig. 6.18 compares the absolute values of the exact )(α+F with respect to the approximated function )(α+aF
fig. 6.18: Comparison of the absolute values of the exact and approximate factorized plus function
In figures 6.19 and 6.20 the relative errors of this approximation )(
)()()(
ααα
α+
++ −=
FFF
e a
assuming A=10, h=1 and A=50,h=1, respectively, have been reported.
fig.6.19: Relative error of the approximate plus solution for A=10
144
fig.6.20: Relative error of the approximate plus solution for A=50
Many diffraction problems, e.g. the one of the considered example, yield solutions that, suitably normalized, are independent on the propagation constant k. This suggests to assume this parameter with a significant imaginary part, in order to locate the branch points k± far from the real axis α and to avoid in this way the previously considered
problems. For instance, fig 6.21 shows the relative error )(
)()()(
ααα
α+
++ −=
FFF
e a when
k=1-j has been assumed.
fig.6.21: Relative error of the approximated plus solution for A=50 and for a strong imaginary part of k.
Smaller values of h diminish the accuracy, e.g. see fig.6.22.
fig.6.22: Relative error of the approximate plus solution for A=5 and h=1/10 and for a strong imaginary
part of k.
145
Conversely, by increasing the value of A the accuracy improves, e.g. see fig. 6.23.
fig.6.23: Relative error of the approximated plus solution for A=50 and h=1 and for a strong imaginary part
of k.
1.6.1.3.1 Presence of a continuous and an infinite discrete spectrum Let us consider the use of the Fredholm equation for solving the W-H problem:
(53) 1
1)()()(αα
ααα−
+= −+ FFG
where 1α is defined by eq. (52) of 1.3.2.3 (Example 5) and the kernel is given by:
(54) ]cos[
)(d
eGdj
ττα
τ
= , 22 ατ −= k
The exact solution of this equation is:
(55) 1
111 )()()(
αααα
α−
=−−
−+
+
GGF
where the factorized functions )(α±G have been obtained in 1.3.2.3 (Example 5). The Fredholm equation for this case is:
(56) o
hA
hAiihFihm
jhFG
ααα
παα
−=+ +
−=+ ∑ 1)(),(.
2)()(
/
/
where:
])()(),(,[),(ααααα
−−
===t
GtGmtIftm o
146
with:
]cos[)tan(
]cos[1)( 222/3 b
bebb
ejbembjbjbj
o ττ
αττ
ατ
ατ
ατττ
−
−= -
We define the following matrices of order 2A+1
(57) }]/,/,{},,,,{],,[[2
hAhAihAAxihxmTablej
hM −−=π
(58) }],,,{],[[[ hAAxxGTabletrixDiagonalMaD −= and the vector
(59) }],,,{,1[1
hAAxx
Tables −−
=α
The vector ],[ sMDeLinearSolvu += defines the samples hAiihF /,...,2,1,0),( ±±±=+ . It yields the approximate solution:
(60)
−
+−= +−=
−+ ∑
1
/
/
1 1)(),(.2
)()(αα
απ
αα ihFihmj
hGFhA
hAia
Figures 6.24 and 6.25 show the comparison between the exact solution and the approximate solution with different choice of A and h.
fig. 6.24: Comparison between the exact and the approximate solution with A=10 and h=1
147
fig. 6.25: Comparison between the exact and the approximate solution with A=40 and h=0.25
1.6.1.3.2 Presence of infinite discrete spectrum and absence of continuous
spectrum Let us consider the use of the Fredholm equation for solving the W-H problem :
(61) ]][[
1)()()(i
FFGaαα
ααα−
+= −+
where 1α is defined by eq. (49) of 1.3.2.2 (Example 4) and the kernel is given by:
22
+−=
cbikinπα 0]Im[....,2,1 <= aii α ,
(62) )](sin[)sin()sin()(
cbcbG
+=
ττττα , , 22 ατ −= k
The exact solution of this equation is:
(63) 1
111 )()()(
αααα
α−
=−−
−+
+
GGF
where the factorized functions )(α±G have been obtained in 1.3.2.2 (Example 4). The Fredholm equation for this case is:
(64) ]][[
1)(),(.2
)()(/
/ iihFihm
jhFG
a
hA
hAi ααα
παα
−=+ +
−=+ ∑
where:
148
])()(),(,[),(ααααα
−−
===t
GtGmtIftm o
with: (65)
[ ] [ ] [ ][ ] [ ]
[ ]( )
[ ] [ ] [ ] [ ]222222
2222
22
22
22
2222
22
222222
sin sin csc cot csc
csc cos
sin csc cos)(
23 αα
αααα
α
αα
αααα
αααααα
−
−
−−−
+−
−+
+
−−−
−+
+−
−−−−=
kckbk
kakaak
ka
kkakbb
kkbkakccmo
Let us introduce the following matrices of order 2A+1
(66) }]/,/,{},,,,{],,[[2
hAhAihAAxihxmTablej
hM −−=π
(67) }],,,{],[[[ hAAxxGTabletrixDiagonalMaD −= and the vector:
(68) }],,,{,1[1
hAAxx
Tables −−
=α
The vector ],[ sMDeLinearSolvu += defines the samples hAiihF /,...,2,1,0),( ±±±=+ . It yields the approximate solution:
(69)
−
+−= +−=
−+ ∑ ]][[
1)(),(.2
)()(/
/
1
iihFihm
jhGF
a
hA
hAia αα
απ
αα
Figures 6.26 and 6.27 report the comparison between the exact and the approximate solution for different choices of ]][[iaα :
149
fig. 6.26: Comparison between the exact and the approximate solution for an evanescent incident mode.
fig.6.27: Comparison between the exact and the approximate solution for a propagating incident mode
1.6.1.3.3 Some expedients to improve the convergence of the numerical solution in
the the plane−α The integral equation (49) could involve a kernel that is not a compact operator. In this case we are not dealing with Fredholm integral equation of second kind. In the next section we will proof that if 1( ) and ( ) exists as G Gα α α− →∞ on the real axis as α → ±∞ , the kernel of the integral equation (49) is of second kind and it allows to all the powerful apparatus developed for numerically solving these equations. Generally the numerical solution of Fredholm integral equation of second kind by using simple numerical schemes of solutions is very efficient. For instance theoretical considerations ( Kantorovich L.V. , V. I. Krylov , Approximate methods of higher analysis, Interscience, New York, Noordhoff, 1967, Gronigen ) ascertain that diminishing h and increasing A assure that the numerical solution converges to the exact solution of the integral equation. However many times we are dealing with equations where the critical points constituted by the singularity oα of the source and/or the structural singularities of the kernel are near the integration lines. These circumstances
150
could damage the convergence of the numerical solution and procedures that overcome these problems are indicated in the literature (See for example Kantorovich L.V. , V. I. Krylov , Approximate methods of higher analysis, Interscience, New York, Noordhoff, 1967, Gronigen ). However another significant expedient to improve the numerical solution of a Fredholm equation related to a W-H equation is to warp the contour path constituted by the real α − axis into a modified line that is located enough far from the critical points. This expedient was already used with success by this authors to factorize the matrices of order four involved in the diffraction by impenetrable wedges problems {SIAM and IEEE). To this end let’s indicate with ( ),y yλ −∞ < < ∞ a line on the planeα − located enough far from the critical points of the Fredholm equations and such that there are not critical points. For instance in wedge problems {SIAM and IEEE) the critical point includes branch points k± and the pole oα that fall down on the real axis for vanishing values of the loss in the free space. So to avoid critical points near the integration line we can introduce as integration contour in the planeα − the line λ defined by
(1.5) 4( ) ,j
y k e y yπ
α = −∞ < < ∞ (fig.3).
This line was that used in wedge problems (IEEE).
fig.3 Warping of the real axis on the line 4( ) ,j
y k e y yπ
λ = −∞ < < ∞ in wedge problems.
However other contours are possible . These authors also used integration contours defined by:
151
(1.6) 2
arctan( ) [ ]yy k y jp qx
α = ++
, y−∞ < < ∞
where p and q are positive parameters,
contours defined by:
(1.7) ( ) cos ( )y k w yα = −
where after the introduction of the Gudermann’s function 1( ) cos sgn( )cosh
gd x arc xx
= ,
it is:
1( ) cos[ ( ) ]2 2
w y k gd y j yπ= − − + − − , y−∞ < < ∞
In particular contours defined by
(1.8) ( ) cos( )2
y k j yπα = − − + y−∞ < < ∞
resulted very efficient to factorize matrix kernels involved in wedge problems (IEEE)
If no singularities of ( )G α and 1( )G α− are present in the region between real axis and ( )yα , the warping to the line ( )yα reduces the Fredholm equation (1.34) to the
equation:
(1.9) [ ] 11 ( ( )) ( ( )) ( ( )) ( ( ))( ( ))( ( )) . '( ) ,
2 ( ) ( ) ( ) o
G t x G y F t x G y RG yF y t x dxj t x y y
x
α αααπ α α α
−− +∞ ++ −∞
− ⋅ ⋅= − +
− −
−∞ < < ∞
∫
where ( ) ( )t x xα=
The discretized version of (1.44) is:
(1.10) /
/( ( ) ( ) . ( ( ), ( )) ( ( )) '( )
2 ( )
A h
i A h o
h RG y F m y t h i F t h i t hij y
α α απ α α+ +
=−
+ =−∑
and it yields the system :
152
(1.11) /
/( ( ) ( ) . ( ( ), ( )) ( ( )) '( )
2 ( )
A h
i A h o
h RG h r F h r m h r t h i F t h i t hij h r
α απ α α+ +
=−
+ =−∑
0, 1, 2,... Arh
= ± ± ±
This system of order 12 +hA yields the samples ( ( )), 0, 1, 2,..., /F t hi i A h+ = ± ± ± . With
these sample we obtain yields the approximate solution in the planeα − of the W-H unknowns:
(1.12)/
1
/ 0
( ) ( ) . ( , ( )) ( ) '( )2
A h
i A h
RhF G m t hi F hi t hij
α α απ α α
−+ +
=−
= − + −
∑
(1.13) /
/ 0
( ) . ( , ( )) ( ( )) '( )2
A h
i A h
h RF m t hi F t h i t hij
α απ α α− +
=−
= − +−∑
Analytical continuation outside the integration line
Eq.s (1.47) and (1.48) provide the representations of ( )F α+ and ( )F α− in all the planeα − through an analytical continuation of the numerical solution obtained on the
point of the integration line of the Fredholm equation. We must be careful in presence of branch lines in this plane. In fact this analytical continuation provides correct results only for the principal branch of ( )F α+ and ( )F α− . For instance in the wedge problems it is convenient to introduce Sommerfeld functions to express diffracted field. These functions are defined in the w-plane where cosk wα = − . To relate Sommerfeld functions to the W-H solutions everywhere we also need the introduction of other branch of ( )F α+ beside the principal one. It requires suitable analytical continuations in the w plane− that are described in sect. 1.4.4 (ICEAA, SIAM, IEEE AP). For instance the W-H equation for wedge problem in the w plane− assumes the form (SIAM):
)(ˆ)(ˆ)(ˆ Φ−−= ++ wYwXwG
where the plus function are even functions of w .
Taking into account the previous property of the plus functions, it is possible to eliminate the unknown )(ˆ wY+ obtaining the relationship:
)(ˆ)(ˆ)2(ˆ)2(ˆ 1 wXwGwGwX +−
+ Φ−−=Φ+
153
Starting from the values of ˆ ( )X w+ known the images of the proper sheet , this equation allows to evaluate for every value of w (IEEE)
Other aspects to analyze are constituted from the following facts: a) the approximate solutions (1.44) introduces spurious poles ; b) in some case it is important to evaluate residues of ( )F α+ and ( )F α− in points located in structural singularities defined by the zeroes of det[ ( )]G α and 1det[ ( )]G α− .
For what concerns the fact a), we observe in the exact representation of ( )F α+ :
(1.14) [ ]11( ) ( ) ( ) ( )1( ) . ( )
2 o
G G G t F t RF dt Gj t
α αα α
π α α α
−+∞ + −
+ −∞
⋅ −= − + ⋅
− −∫ Im[ ] 0oα <
there is a compensation of the minus structural singularities (poles or branch points located in the upper half-plane Im[ ] 0α > ) in the second member. To show it, let’s suppose the presence of the zero cα of det[ ( )]G α in Im[ ] 0α > . Looking at the equation:
(1.15) [ ]11( ) ( ) ( ) ( )1( ) . ( )
2 o
G G G t F t RF dt Gj t
α αα α
π α α α
−+∞ + −
+ −∞
⋅ −= − + ⋅
− −∫ Im[ ] 0oα < ,
the residue in cα of the first term of the second member is expressed by:
(1.16) 1 1( ) ( ) ( )1 1Res ( ) Res ( )2 2c c cc c
G t F t F tR G dt G dtj t j tα α α
α απ α π α
+∞ +∞− −+ −
−∞ −∞
− = − ⋅ = ⋅ − −∫ ∫
Taking into account that the ( )F t− present in the half plane Im[ ] 0α < only the pole oα with residue R, the previous equation yields:
(1.17)
1 1
1
( ) ( ) ( )1 1Res ( ) Res ( )2 2
Res ( )
c c c
c
c c
c o
G t F t F tR G dt G dtj t j t
RG
α α α
α
α απ α π α
αα α
+∞ +∞− −+ −
−∞ −∞
−
− = − ⋅ = ⋅ − −
= − ⋅ −
∫ ∫
Since the second term presents a residue 1Res ( )c c o
RGα
αα α
− ⋅ − in the pole cα that
compensates the previous one, it follows that the exact expression of ( )F α+ given by (1.50) does not present the spurious pole cα located in the upper half-plane Im[ ] 0α > )
154
Consequently really ( )F α+ is a plus function.
The same reasoning can be used when a branch line is present in 1( )G α− in the upper half-plane Im[ ] 0α > . In fact the jump on the two lips in the first term of the second member of (1.49) yields:
(1.18) 1
1 1( ) ( ) ( ) ( )1 1( ) ( )2 2 o
G G t F t F t Rdt G dt Gj t j t
α α απ α π α α α
−+∞ +∞− −+ −
−∞ −∞
⋅ ⋅ ∆ = ∆ = −∆ − − − ∫ ∫ .
It compensate the jump in the second term: 1( )o
RG αα α
− ∆ −.
The previous compensations for offending minus singularities present in the exact representations derive from the relationship
(1.19) ( ) ( )12 c c o
G t F t Rdtj tπ α α α
+∞+
−∞⋅ = −
− −∫
that holds if cα is located in the upper half-plane Im[ ] 0α > .
Conversely in the approximate representation:
(1.20) /
/
( ) ( )2
i A h
i A h c c o
G h i F h ih Rj h iπ α α α
=+
=−
⋅ ≈ −− −∑
the presence of the sum instead of the integral does not assure this compensation. It follows the offending presence of minus singularities in plus functions ( )F α+ . Consequently the quantities:
(1.21) /
/
( ) ( )2
i A h
i A h o
G h i F h ih Rj h iπ α α α
=+
=−
⋅ ≈ −− −∑
for every value of α in Im[ ] 0α > represents an accuracy indicator of the presence of offending singularities.
Similar considerations can be done for the minus function ( )F α− .
155
For the case b), generally the residues of the poles of ( )F α+ (or ( )F α− ) have an important physical meaning. For instance the residue of ( )F α+ in the pole iα of
1( )G α− represent the excitation coefficients of the mode present in the plus region and having the propagation constant iα . Fortunately since we have:
(1.22) 1Res[ ( )] Res[ ( )] ( )i i iF G Fα α α αα α α−
+ = = −=
and we can evaluate exactly 1Res[ ( )]i
G α αα−= , the accuracy of the evaluation of
Res[ ( )]i
F α αα+ = depends on the accuracy of the evaluation of ( )iF α− through the approximate expression given by (1.48). Since iα is a point located in the regular region of ( )F α− there are not problem for having a good evaluation of ( )iF α− .
Compactness of the kernel 1( ) ( )G G u
uα
α
− −−
1
With the conditions that ( )G α and 1( )G α− are bounded on the real axis as α → ±∞ , in
the following we will state that the kernel 1( ) ( )G G u
uα
α
− −−
1 of the integral equations
1.12, 1.16 and 1.17 is compact. Firstly we rewrite:
(1.23) 1
1( ) ( ) ( ) ( )( )G G u G u GGu uα αα
α α
−−− −
=− −
1
Since ( )G α and 1( )G α− are bounded, an important property of the functional analysis
reduce to the proof the compactness of [ ]( ) ( )G u Gu
αα
−−
The discussion of this kernel is considered in [62]. In this reference the compactness of this kernel has been studied in a general contest. Concerning our applications it is sufficient to show that [63]:
(1.24) [ ] 2( ) ( )rs rsG u G
du du
αα
α∞ ∞
−∞ −∞
−< ∞
−∫ ∫
where ( )rsG u are components of the matrix ( )G u . Taking into account that the difficulty to proof (1.19) arises from the presence of an infinite interval of integration we limit our self to show that the contribute of
156
(1.25) [ ] 2( ) ( )rs rs
M M
G u Gdu d
uα
αα
∞ ∞ −−∫ ∫ and [ ] 2
( ) ( )M rs rs
M M
G u Gdu d
uα
αα
∞
−
−−∫ ∫
where M is a sufficiently large number are bounded. Let’s suppose that in the interval M x≤ < ∞ , ( )rsG u is represented by the convergent series
(1.26)1 2
0 02
1
1( ) ...i
rs rs rsrs rs rs i
i
A A AG u A Au u u u u uν ν ν
∞
=
= + + + = + ∑ ,
(where 0ν ≥ ) It follows:
(1.27) 1
( ) ( ) ( ) ( ) i i irs rs rs rs rs
i ii
G u G G u G u A u uu u u u u u u
ν ν
ν ν
αα
+ +∞> < > <
+ +=> < > < > <
− − −= =
− − −∑
where u> (u< )stands the greater (smaller) between u and α The previous representation yields:
(1.28) 2 *
1 1
( ) ( ) ( )i j i i j jrs rs rs rs
i i j ji j
G u G A A u u u uu u u u u u u u u
ν ν ν ν
ν ν ν ν
αα
+ + + +∞ ∞> < > <
+ + + += = > < > < > < > <
− − −=
− − −∑∑
applying the mean value theorem we have :
(1.29) 1( )i i
iu u i uu u
ν ννν
+ ++ −> <
>> <
−≤ +
− ( 1,2...i = )
where the equal holds when 0 and i=1ν = Taking into account that (1.30)
* *
2 2 2 2
** *
2 2 2 2
( ) ( )( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( )
i j j irs rs rs rs
i j j i
i j i jrs rs rs rs
i j i j
A A A Ai j j iu u u u
A A A Ai j i ju u u u
ν ν
ν ν
ν ν ν ν
ν ν ν ν
+ + + +< > < >
+ + + +< > < >
+ + + + + =
= + + + + +
157
is real , we have:
(1.31)
*
1 1
*
2 21 1
( )
( )( ) ( )
i j i i j jrs rs
i i j ji j
i jrs rs
i ji j
A A u u u uu u u u u u u u
A Ai ju u
ν ν ν ν
ν ν ν ν
νν ν
+ + + +∞ ∞> < > <
+ + + += = > < > < > < > <
∞ ∞
+ += = < >
− −<
− −
< + +
∑∑
∑∑
It follows:
(1.32)
2 *
2 21 1
*
21 1
( ) ( ) ( )( ) ( )
( )( ) ( )2 1
i jrs rs rs rs
i jM M M Mi j
i jrs rs
i ji j
G u G A Adu d i j du duu u u
A Ai ji j M
ν
ν
α α ν να
ν νν
∞ ∞∞ ∞ ∞ ∞
< >+ += = < >
∞ ∞
+ += =
−< + + =
−
+ +=
+ + −
∑∑∫ ∫ ∫ ∫
∑∑
Now we show that the last double series is convergent. In fact we have: (1.33)
22
,
( ) ( ) ( ) ( )*
1 1 ,
*
2 21 1
( ) ( ) '( )
( )
( )1 ( )( )
rs rsrs
x M M
i i j ji jrs rs
i j u M M
i jrs rs
i ji j
G u G G Mu
u uA Au u
A Ai jM M
α
ν ν ν ν
α
ν
αα
α αα α
ν ν
= =
− + − + − + − +∞ ∞
= = = =
∞ ∞
+ += =
−= =
−
− −= = − −
= + +
∑∑
∑∑
Since:
(1.34) 1 12 1i jν
≤+ + −
it follows:
(1.35)
2 *
21 1
*22
2 21 1
( ) ( ) ( )( ) ( )2 1
( )1 ( )( ) '( )
i jrs rs rs rs
i jM Mi j
i jrs rs
rsi ji j
G u G A Ai jdu du i j M
A Ai j M G MM M
ν
ν
α ν ναα ν
ν ν
∞ ∞∞ ∞
+ += =
∞ ∞
+ += =
− + +< <
− + + −
< + + =
∑∑∫ ∫
∑∑
This result proofs that the integral [ ] 2( ) ( )rs rs
M M
G u Gdu d
uα
αα
∞ ∞ −−∫ ∫ is bounded.
158
For what it concerns the integral [ ] 2( ) ( )M rs rs
M M
G u Gdu d
uα
αα
∞
−
−−∫ ∫ ( ,M u M Mα− ≤ ≤ ≥ )
we have: (1.36)
[ ] 22
2
22
2 2 2
22 2
( ) ( ) 1 ( ) ( )
( ) ( )1 ( ) 2 ( )
( )1 ( ) 2 ( )
M Mrs rsrs rsM M M M
M M Mrs rsrs rsM M M M M M
M rsrs rsM M
G u Gdu d G u G du d
u
G GG u du d G u du d du d
GG u du d G u du
αα α α
α α
α αα α α
α α αα
αα α
∞ ∞
− −
∞ ∞ ∞
− − −
∞
− −
− < − < −
< + + <
< +
∫ ∫ ∫ ∫
∫ ∫ ∫ ∫ ∫ ∫
∫ ∫M
M Mdα
∞ ∫ ∫
Since the integrals in u are convergent and ( )rsG α is bounded as α →∞ ,
We obtain that [ ] 2( ) ( )M rs rs
M M
G x Gdx d
xα
αα
∞
−
−−∫ ∫ is bounded too.
This proof extends also to the case where ( )rsG x is represented by the series:
(177) 0 1 22
1 1( ) ( ) ( ) ( ) ...rs rs rs rsG x G x G x G xx xν ν= + + +
where 0ν > and every ( )l
rsG x (l=1,2..) can be expanded in the series:
(178) 1 2
02( ) ...
l ll l rs rsrs rs
A AG x Ax x
= + + +
We have :
(179)
222
0
2 2 2 2
11 10 0 0 0
( ) ( ) 1 ( )
1 1( ) '( )
lrs rsrslM M
l
l lrs rs M Ml l l l
l l l l
G x G ddx d M G Mx dM M
l lG M G M G GM M M M
ν
ν ν ν ν
α αα
ν ν
∞∞ ∞
=
∞ ∞ ∞ ∞
+ += = = =
− < < −
< + < +
∑∫ ∫
∑ ∑ ∑ ∑
where 1 and M MG G are chosen so that:
2( )l
M rsG G M> , 2
1 '( )lM rsG G M>
for every value of l.
159
Since the two series in the last member converge, also in this case it follows the
compactness of the operator ( ) ( )rs rsG x Gx
αα
−−
It follows that (102)bis holds. Sometimes the series (22) is not correct since we have essential singularities at infinity. For instance it happens dealing with kernel having on the integration line the asymptotic representation:
20 1 2( ) ...d u d urs rs rs rsG u A A e A e− −= + + +
where d is a positive parameter . In these case for the presence of functions vanishing
exponentially, the compactness of ( ) ( )rs rsG x Gx
αα
−−
can be proved more easily since we
have (ad_comp.nb):
2 21 1( ) ( ) ...
d u d d u drs rs
rs rsG u G e e e eA A
u u u
α ααα α α
− − − −− − −= + +
− − −
Another extension concerns with the case where ( )rsG x is represented by the series:
(177) 0 1 22
1 1( ) ( ) ( ) ( ) ...rs rs rs rsG x G x G x G xx xν ν= + + +
where 0ν > and every ( )l
rsG x (l=1,2..) can be expanded in the series:
(178) 1 2
02( ) ...
l ll l rs rsrs rs
A AG x Ax x
= + + +
We have :
(179)
222
0
2 2 2 2
11 10 0 0 0
( ) ( ) 1 ( )
1 1( ) '( )
lrs rsrslM M
l
l lrs rs M Ml l l l
l l l l
G x G ddx d M G Mx dM M
l lG M G M G GM M M M
ν
ν ν ν ν
α αα
ν ν
∞∞ ∞
=
∞ ∞ ∞ ∞
+ += = = =
− < < −
< + < +
∑∫ ∫
∑ ∑ ∑ ∑
where 1 and M MG G are chosen so that:
2( )l
M rsG G M> , 2
1 '( )lM rsG G M>
for every value of l.
160
Since the two series in the last member converge, also in this case it follows the
compactness of the operator ( ) ( )rs rsG x Gx
αα
−−
F-kernels
In the previous section we shown that we are dealing equations with Fredholm equation
of second kind provided that both 1( ) ( )and 0G Gα α
α α
−
⇒ on the real axis as
α → ±∞ .
In the following we call F-kernels the kernels that satisfy these conditions. The possibility to normalize the kernel and eventually the unknown in the original W-H equations, can in general reduce an arbitrary W-H equation to a Fredholm equation having F-kernels. In the presence of F-kernel, we experienced a significant improving of the numerical solution of the Fredholm equations so we suggest to deal always with Fredholm equations with F-kernel .
In the following we will call structural singularities the singularities present in ( )G α and 1( )G α− .
For example the kernel of sect.1.6.2.2.1 becomes a F-kernel by multiplying it for the well known factorizable function 2 2kτ α= − . It yields the factorization of the k-kernel:
(76) ( )cos[ ]
j degd
τ
ατ
=
The singularities of this kernel are:
branch points : ,kα = ±
poles of ( )g α :
(77) 2
2 ( 1/ 2) 1, 2..nnk n
dπα − ± = ± − =
161
Similarly the kernel of the sect. 1.6.2.2.2 multiplied by 2 2kτ α= − yields the k- kernel:
(78) sin( )sin( )( ) ,sin[ ]b cG b c a
aτ τα
τ= + =
In this case the modified kernel presents continuous spectrum for the presence of the branch points kα = ± . However there it is convenience in the Fredholm framework since
( )G α and its inverse are bounded.
The singularities of this kernel are:
branch points : ,kα = ±
poles of ( )G α :
(79)2
2 1, 2..annk naπα ± = ± − =
poles of 1( )G α− or zeroes of ( )G α :
(80)2
2 1, 2..bnnk nbπα ± = ± − =
22 1, 2..cn
nk ncπα ± = ± − =
Let’s observe that in this two examples the singularities of the kernel are located in the standard branch lines b+ and b− (fig.2.3) so that k± and the all the nα± are very near the real axis for vanishing values of the imaginary part of k .
Another significant expedient to improve the numerical solution of a Fredholm equation related to a W-H equation is to warp the contour path constituted by the real α − axis into a modified line far from the singularities and/or the zeroes of the kernels. Let’s indicate with ( ),y yλ −∞ < < ∞ a line on the planeα − located enough far from the poles of ( )G α and 1( )G α− such that there are not poles of the kernel between the real axis and this line. For instance taking into account the location of the singularities in the two previous examples (equations (76) and (77), the straight line
162
(81) 4( ) ,j
y e y yπ
λ = −∞ < < ∞ (fig.6.28) satisfies this condition.
Fig. 6.29 considers the case where it has been assumed
(82) ( ) Re[ ]( arctan )y k y j yλ = +
Fig.6.29bis illustrates the case where it has been assumed:
2
arctan( ) [ ](1 )
xx k x jp x
λ = ++
By changing the parameter p , the integration path can be assumed near the original real axes with the exception of the branch points and physical source poles.
The deformation of the real axis to the line ( ) ( )y yα λ= reduces the Fredholm equation (70) to the equation:
(83)[ ] 11 ( ( )) ( ( )) ( ( )) ( ( ))( ( ))( ( )) . '( ) ,
2 ( ) ( ) ( )o
o
G t x G y F t x G y FG yF y t x dxj t x y y
t
α αααπ α α α
−− +∞ ++ −∞
−= − +
− −
−∞ < < ∞
∫
where ( ) ( )t x xλ=
163
[ ] 11 ( ( )) ( ( )) ( ( )) ( ( ))( ( ))( ( )) . '( ) ,2 ( ) ( ) ( )
o
o
G t x G y F t x G y FG yF y t x dxj t x y y
y
α αααπ α α α
−− +∞ ++ −∞
−= − +
− −
−∞ < < ∞
∫
The sampled version of (83) is:
(84) /
/( ( ) ( ) . ( ( ), ( )) ( ( )) '( )
2 ( )
A ho
i A h o
FhG y F m y t h i F t h i t hij y
α α απ α α+ +
=−
+ =−∑
where:
(85) ( ) ( ))( , ) [ , '( ), ]G G tm t If t Gt
αα α αα−
= ==−
We define the following matrices of order 2A+1
164
(86) [ [ ( ), ( )],{ , , , },{ , / , / }]2
hM Table m y t h i y A A h i A h A hj
απ
= − −
(87) [ [ [ ( )],{ , , , }]D DiagonalMatrix Table G y y A A hα= − and the vector
(88) 1
[ ,{ , , , }]( )
oFs Table y A A hyα α
= −−
The vector ],[ sMDeLinearSolvu += defines the samples ( ( )), 0, 1, 2,..., /F t hi i A h+ = ± ± ± . It yields the approximate solution:
(89) /
1
/ 0
( ( )) ( ( )) . ( ( ), ( )) ( ( )) '( )2 ( )
A ho
ai A h
FhF y G y m y t h i F t hi t hij y
α α απ α α
−+ +
=−
= − + −
∑
or in the planeα −
(90) /
1
/ 0
( ) ( ) . ( , ( )) ( ) '( )2
A ho
ai A h
FhF G m t hi F hi t hij
α α απ α α
−+ +
=−
= − + −
∑
(91) /
/ 0
( ) . ( , ( )) ( ( )) '( )2
A ho
ai A h
FhF m t hi F t hi t hij
α απ α α− +
=−
= − +−∑
In the following we will apply the following considerations to the kernels (76) and (78) Fredholm factorization of the kernel (76)
The Fredholm factorization of the kernel 2 2, , 0cos[ ]
j de k dd
τ
τ ατ τ
= − > has been
considered in sect.1.6.2.2.1. Figures 6.24 and 6.25 show results not very accurate so to improve them it is suggested to use the techniques indicated in the previous section. Firstly we normalize the kernel yielding the F-kernel (76):
(92) 2 2
2 2( ) ,
cos[ ]
j k degk d
α
αα
−
=−
In the following we will obtain numerical solution of the Fredholm equation in three cases. With respect the figures 6.24 and 6.25 we introduced a free propagation constant k
with very low imaginary part: 82 (1 10 )k jπλ
−= − . It involves some singularities of the
kernel very near the real axis. However, as discussed in the previous section we will
165
overcome this unpleasant circumstance by warping the original line constituted by the real axis in another line far from the singularities. To estimate the accuracy of the approximate solution a physical important parameters is the residue of the minus function ( )F α− in the pole (1) 1dα α− = − (see sect. 1.3.2.3). We have: (93)
1 1 11 1Re [ ( )] Re [ ( )] ( ) Re [ ( )] ( )as F s g F s g F
α α αα α α α α− + +− − −
= − ≈ −
Consequently the accuracy of the numerical solution can be obtained by comparing
1( )aF α+ − with the exact value 1( )F α+ − . We obtain:
(94) ( ) ( ))( , ) [ , ( ), ]og g tm t If t m
tαα α αα−
= ==−
where:
(95) 2 2 2 2
( ) '( )cos
ojdm g
k k dαα α
α α= = −
− −
Case a: the numerical solution is considered on the original line constituted by the real axis. (miste_fredholm5bisa1.nb) Fig. 6.30 illustrates the obtained results. Using a F-kernel improves considerably the accuracy (compare with fig. 6.24 and 6.25). However we have:
1( ) 0.03738 0.00291aF jα+ − = − −
1( ) 0.04823 0.01040F jα+ − = − +
166
Case b: the numerical solution is considered on line 4( ) ,j
y e y yπ
λ = −∞ < < ∞ (fig.6.28) (miste_fredholm5bisa2a.nb) Fig. 6.31 illustrates the obtained results. Using a F-kernel and solving on the line
4( ) ,j
y e y yπ
λ = −∞ < < ∞ improves the accuracy (compare with fig.6.30). Now we have:
1( ) -0.0427751 +j 0.017618 aF α+ − =
1( ) 0.04823 0.01040F jα+ − = − +
167
Case c: the numerical solution is considered on line ( ) Re[ ]( arctan )y k y j yλ = + (fig.6.29) (miste_fredholm5bisa3a.nb) Fig. 6.32 illustrates the obtained results. Using a F-kernel and solving on the line
( ) Re[ ]( arctan )y k y j yλ = + improves the accuracy (compare with fig.6.30). Now we have:
1( ) -0.0480818 + j 0.0130597 aF α+ − =
1( ) 0.04823 0.01040F jα+ − = − +
Fredholm factorization of the kernel (78)
The Fredholm factorization of the kernel sin( )sin( )( ) ,sin[ ]b cG b c a
aτ τατ τ
= + = has been
considered in sect.1.6.2.2.2. Figures 6.26 and 6.27 show results not very accurate so to improve them it is suggested to use the techniques indicated in the previous section. Firstly we normalize the kernel yielding the F-kernel (78):
(96) sin( )sin( )( ) ,sin[ ]b cG b c a
aτ τατ τ
= + =
168
In the following we will obtain numerical solution of the Fredholm equation in three cases. With respect the figures 6.26 and 6.27 we introduced a free propagation constant k
with very low imaginary part: 82 (1 10 )k jπλ
−= − . It involves some singularities of the
kernel very near the real axis. However, as discussed in the previous section we will overcome this unpleasant circumstance by warping the original line constituted by the real axis in another line far from the singularities. To estimate the accuracy of the approximate solution a physical important parameters is the residue of the minus function ( )F α− in the pole 1 1aα α− = − (see sect. 1.3.2.2). We have: (97)
1 1 11 1Re [ ( )] Re [ ( )] ( ) Re [ ( )] ( )as F s g F s g F
α α αα α α α α− + +− − −
= − ≈ −
Consequently the accuracy of the numerical solution can be obtained by comparing
1( )aF α+ − with the exact value 1( )F α+ − . Another parameter that estimate the accuracy is the residue of the minus function ( )F α− in the poles 1bα and 1cα (see sect. 1.3.2.2). We have: (98)
1 1 1
1 11 1Re [ ( )] Re [ ( )] ( ) Re [ ( )] ( )
b b bb a bs F s G F s G F
α α αα α α α α− −
+ − −= ≈
(99) 1 1 1
1 11 1Re [ ( )] Re [ ( )] ( ) Re [ ( )] ( )
c c cc a cs F s G F s G F
α α αα α α α α− −
+ − −= ≈
Consequently the accuracy of the numerical solution can be obtained by comparing , 1( )a b cF α− with the exact value , 1( )b cF α− .
We obtain:
(100) ( ) ( ))( , ) [ , ( ), ]oG G tm t If t m
tαα α αα−
= ==−
(101) ( ) '( )om Gα α= The expression of ( )om α is not reported here. Case a: the numerical solution is considered on the original line constituted by the real axis. (meromorfefredholm5_a1.nb) Like the case of not F-kernel (fig. 6.26) and (6,27) the Fredholm solution is not sufficiently accurate so it is not reported here
169
Case b: the numerical solution is considered on line 4( ) ,j
y e y yπ
λ = −∞ < < ∞ (fig.6.28) (meromorfefredholm5_a2.nb) Fig. 6.33 illustrates the obtained results. Using a F-kernel and solving on the line
4( ) ,j
y e y yπ
λ = −∞ < < ∞ considerably improves the accuracy (compare with fig.6.26) and (6,27).
Observe the peaks for negative α in the approximate solution. They are due to the poles
1bα− and 1cα− very near to the real axis that are spuriously present in the approximate solution ( )aF α+ Now we have:
1( ) -0.0766365-j0.134256 aF α+ − =
1( ) -0.0735487-j0.131697 a aF α+ − =
1( ) -0.300055-j0.0625272 bF α− =
1( ) -0.294696-j 0.0644443 a bF α− =
1( ) -0.58009-j0.053066 cF α− =
1( ) -0.582995-j0.0543879 a cF α− = Case c: the numerical solution is considered on line ( ) Re[ ]( arctan )y k y j yλ = + (fig.6.29) (meromorfefredholm5_a3.nb)
170
Fig. 6.34 illustrates the obtained results. Using a F-kernel and solving on the line
( ) Re[ ]( arctan )y k y j yλ = + (compare with fig.6.33). Now we have:
1( ) -0.0766365-j0.134256 aF α+ − =
1( ) -0.0655048-j0.119111 a aF α+ − =
1( ) -0.300055-j0.0625272 bF α− =
1( ) -0.289664-j0.0462318 a bF α− =
1( ) -0.58009-j0.053066 cF α− =
1( ) -0.577487-j0.0543803 a cF α− =
821 (1 10 ), ( )
( ) 20, 0.5
1.1 , 1.32 2
a
k j F
F A h
b c a b c
πλ αλ
α αλ λ
−+
+
= = −
= =
= = = +
1.6.2 Numerical solution of the W-H equations in the planew − In order to solve the Fredholm integral equation by quadratures, sometimes the kernel presents a more suitable behavior in the w-plane defined by cosk wα = − . Let us consider in fact the W-H equation involved in wedge problems:
171
(102)o
oFFFG
ααααα
−+= −+ )()()(
If )(α+F is a standard plus functions ( 0]Im[ <oα ) it yields the following Fredholm integral equation:
(103) [ ]0]Im[,
)()()(.
21)()( <
−=
−−
+ ∫∞+
∞−
++ o
o
oFdx
xxFGxG
jFG α
αααα
παα
fig.6.35: Deformation of the contour path It is convenient to deform the contour path constituted by the real axis into the straight line αλ reported in fig. 6.28. If there are no poles of )(αG and )(α+F in the shaded region, it yields:
(104)[ ] [ ]
∫∫ −−
=−
− +∞+
∞−
+
αλ αα
παα
πdx
xxFGxG
jdx
xxFGxG
j)()()(
.2
1)()()(.
21
In order to facilitate the theoretic function manipulations, that may be difficult to handle in the α -plane, the angular w complex plane have been introduced: (105) wo cosτα −= 'cos wx oτ−= as it has been considered in sec. 1.2.12.2. In the w-plane wλ and wr are the images of the line αλ and of the real axis of the plane−α , respectively.
172
fig.6.36: Integration paths in the w-plane
Furthermore, (106) ooooo w ϕττα coscos −=−= where oow ϕ−= , πϕπ << o2/ . In the w-plane the Fredholm equation becomes: where oow ϕ−= , πϕπ << o2/ . In the w-plane the Fredholm equation becomes:
(107) [ ]
,coscos
''sincos'cos
)'(ˆ)(ˆ)'(ˆ
21)(ˆ)(ˆ
ooo
oo
oo wwFdww
wwwFwGwG
jwFwG
w τττ
ττπ λ −−=
+−−
+ ∫ ++
By introducing
(108) ujw +−=2
' π
it yields: jdudw =' ujw sinh'cos = uw cosh'sin −= (109) )()'(ˆ uHwG = )()(ˆ uYwF =+
[ ] ,coscos
)cosh(cossinh
)( )(ˆ)(2
1)(ˆ)(ˆooo
o
wwFjduu
wujuYwGuH
jwFwG
ττπ −−=−
+−−
+ ∫∞−
∞++
or
173
(110) [ ] ,coscos
coshcossinh
)( )(ˆ)(2
1)(ˆ)(ˆooo
o
wwFjduu
wujuYwGuH
jwFwG
ττπ −−=
+−−
+ ∫∞+
∞−+
It provides the representation:
(11) [ ] ,coscos
)(ˆcosh
cossinh)( )(ˆ)(
2)(ˆ
)(ˆ11
ooo
o
wwFwG
jduuwujuYwGuH
jwG
wFττπ −
−+−
−−=
−∞+
∞−
−
+ ∫
Let us now deduce the equations valid for the samples )( ihY . By putting tjw +−=2π
in the equation
(112) [ ] ,coscos
coshcossinh
)( )(ˆ)(2
1)(ˆ)(ˆooo
o
wwFjduu
wujuYwGuH
jwFwG
ττπ −−=
+−−
+ ∫∞+
∞−+
it yields:
(113) [ ] ,)cossinh(
)(coshsinhsinh
)()(2
1)()(oo
o
wtjF
duuYutu
tHuHj
tYtH−
−=+−−
+ ∫∞+
∞− τπ
or
(114) ,)cossinh(
)(),(2
1)()(oo
o
wtjF
duuYutmj
tYtH−
−=+ ∫∞+
∞− τπ
where
)()(ˆ uYwF =+ [ ] [ ] ]
sinhsinh)()(,cosh
sinhsinh)()(lim,[),(
tutHuHu
tutHuHtuIfutm
+−−
+−−
===
with the limit evaluated as tu = . The sampled form of the previous Fredholm equation is: (115)
hAr
wrhjFihYihrhm
jhrhYrhH
oo
ohA
hAiaa ±±±=
−−=+ ∑
−=
,...,2,1,0,)cossinh(
)(),(2
)()(/
/ τπ
The solution of the previous equation in 12 +hA unknowns ),...,2,1,0(),(
hAiihYa ±±±=
yields the representation:
174
(116) ,cos)sinh(
)()(),(
2)(
)(1/
/
1
ooo
ohA
hAiaa wtj
FtHihYihtmh
jtH
tYττπ −
−−=−
−=
−
∑
that, in the w-plane, yields the approximate plus function (117) )()(ˆ uYwF aa =+ or:
(118) ,)cos(cos
)(ˆ)(]),
2([
2)(ˆ
)(ˆ1/
/
1
oo
ohA
hAiaa ww
FwGihYihwjmh
jwG
wF−
−+−−=−
−=
−
+ ∑ τπ
π
Compactness of the kernel in the w-plane Working in the w-plane the integral equation to consider is constituted by equation (81) and now we are faced with a different kernel. However returning to the half plane we deal with the double integral:
[ ] 2( ) ( )rs rsG x G
dx dxα αλ λ
αα
α−−∫ ∫
where the straight line αλ is indicated in fig.6.28. By using the new variables defined by:
jx u e δ= , jv e δα = where δ is the angle between the real axis and αλ , the double integral becomes
2( ) ( )rs rsG u G v
du dvu v
∞ ∞
−∞ −∞
− −∫ ∫
! !
where: ( ) ( )j
rs rsG u G x e δ=! , ( ) ( )jrs rsG v G e δα=!
Of course if ( )rsG x has the form : 0 1 12
1 1( ) ( ) ( ) ( ) ...rs rs rs rsG x G x G x G xx xν ν= + + +
we have:
(180) 0 1 22
1 1( ) ( ) ( ) ( ) ...rs rs rs rsG u G u G x G uu uν ν= + + +! ! ! !
where:
175
1 2
02( ) ...
l ll l rs rsrs rs
A AG u Au u
= + + +! !! !
and the proof again applies. Example of application Let us consider again the half-plane problem faced in 1.6.1.1. The solution in the w-plane of the Fredholm equation associated to the W-H equation in the range 0<<− wπ yields the result reported in fig 6.30, where the shown exact and approximate solutions are completely identical. Note that even by changing the propagation constant parameter k, the parameter pattern does not change, since it is independent on k.
fig.6.37: Exact and approximate absolute values of the solution
It must be remarked that the representation (86) stands only in points of the w-plane images of the proper sheet (fig. 2.5 of 1.2.12.2). In fact, the regularity of the integral in this representation is guaranteed only in these points. For instance, in the considered
example, the point 2π
−=w (image of the point 0=α located in the proper sheet)
]0),2
([ π+− wjm is regular. Conversely, the point
23π−=w (image of the point 0=α
located in the improper sheet) ]0),2
([ π+− wjm is singular, and it provides a singularity
in )(ˆ wFa + that is not present in the analytical continuation of the exact solution .
The analytical continuation can be accomplished by taking into account that )(ˆ wFa + is an even function and by introducing the difference equation related to the W-H equation. According to the results of section 1.4.4, it yields: (119) )(ˆ)(ˆ)(ˆ)(ˆ 1 ππππ +−−−=− +
−+ wFwGwGwF
176
By this way it is possible to evaluate )(ˆ wFa + for each value of the w plane starting from
the knowledge of )(ˆ wFa + in point of the w plane that are images of the proper sheet. For
instance, the correct value of )23(ˆ π−+aF follows from:
)21(ˆ)
21(ˆ)
23(ˆ)
21(ˆ)
21(ˆ)
23(ˆ)
23(ˆ 11 πππππππ −−−=−−=− +
−+
−+ aaa FGGFGGF
Some expedients to improve the convergence of the solution in the w plane−
As it happens for the numerical solution in the planeα − also in the w plane− a significant expedient to improve the numerical solution of a Fredholm equation related to a W-H equation is to warp the contour path. We used this expedient in the case of presence of only branch points in the previous section by deforming the image wr of the
real axis onto the vertical line wλ with abscissa 2π
−
fig.6.38 important lines in the w-plane
Choosing the line wλ provides very accurate solutions for kernel relevant to diffraction problems (see chapter 3). It does not happen for the kernels involving an infinite discrete spectrum such those considered in equations (76) and (78). In fact generally in these cases the poles nα that in the planeα − are located on the standard branch lines in practice are located on the line wλ as n →∞ . To avoid the presence of singularities near a line we can warp this line by introducing a suitable function ( ),y yλ −∞ < < ∞ that defines the new line . Firstly we observe that broken lines such the line zz indicated in figure are not convenient since they introduce discontinuities in the derivative
177
'( )yλ involved in the mapping '( )dw y dyλ= . We experienced low accuracy for a line
zz constituted by the half lines 34
w π= − ,
4w π= − and a segment with slope 1.
For kernels considered in equation (78), for illustrative purposes we will consider the line defined by
(120) 1( ) ( )2 2
y gd y j yπλ = − + − − y−∞ < < ∞
where ( )gd x is the Gudermann function defined by:
(121) 1( ) cos sgn( )cosh
gd x arc xx
=
fig.6.39: the gundermann line for the Fredholm equation
In the w-plane the considered line is indicated with gd (fig.6.39 )
In general in the w plane− we have:
(122) ˆ ˆ ˆ( ') ( ) ( ')1ˆ ˆ( ) ( ) sin ' ' ,
2 cos ' cos cosw
or
o
G w G w F w FG w F w w dwj w w k wπ α
+
+
− + =− + − −∫
The deformation of the line to the line ( ) ( )y yα λ= reduces the Fredholm equation (70) to the equation:
178
(123) ( )
ˆ ˆ ˆ( ) ( ') ( ')1ˆ ˆ( ) ( ) sin ' ' ,2 cos cos ' cos
oy
o
G w G w F w FG w F w w dwj w w k wλπ α
+
+
− + − =− − −∫
or:
(124) ( )
1ˆ ˆ ˆ( ) ( ) ( , ') ( ') ' ,2 cos
owy
o
FG w F w m w w F w dwj k wλπ α+ ++ =
− −∫
where:
(125) ˆ ˆ( ) ( ')
ˆ( , ') [ ' , '( ), sin ']cos cos 'w
G w G wm w w If w w G w w
w w
− = = −−
By introducing (126) ( )w yλ= , ' ( )w xλ= it yields: ' '( )dw x dxλ= (127)
( )
1ˆ ˆ ˆ( ( )) ( ( )) ( ( ), ( )) ( ( )) '( ) ,2 cos ( )
owy
o
FG w y F w y m y x F x x dxj k yλ
λ λ λ λπ λ α+ ++ =
− −∫
The sampled version is:
(128) /
/
ˆ ˆ ˆ( ( )) ( ( )) . ( ( ), ( )) ( ( )) '( )2 cos ( )
A ho
wi A h o
FhG w y F w y m y hi F hi hij k y
λ λ λ λπ λ α+ +
=−
+ =− −∑
We define the following matrices of order 2A+1
(129) [ [ ( ), ( )] '( ),{ , , , },{ , / , / }]2 w
hM Table m y hi h i y A A h i A h A hj
λ λ λπ
= − −
(130) [ [ [ ( )],{ , , , }]D DiagonalMatrix Table G y y A A hλ= − and the vector
(131) [ ,{ , , , }]cos( ( ))
o
o
Fs Table y A A hk yλ α
= −− −
179
The vector ],[ sMDeLinearSolvu += defines the samples ˆ ( ( )), 0, 1, 2,..., /F hi i A hλ+ = ± ± ± . It yields the approximate solution:
(132) /
1
/ 0
ˆˆ ˆ( ) ( ) . [ , ( )] ( ( ) '( )2 cos( )
A ho
a wi A h
FhF w G w m w hi F hi h ij k w
λ λ λπ α
−+ +
=−
= − + − −
∑
or in the planeα −
(133) /
1
/ 0
ˆ( ) ( ) . ( , cos( ( )) ( ( ) '( )2
A ho
ai A h
FhF G m k hi F hi h ij
α α α λ λ λπ α α
−+ +
=−
= − − + −
∑
(134) /
/ 0
ˆ( ) . ( , cos ( )) ( ( ) '( )2
A ho
ai A h
FhF m k hi F hi h ij
α α λ λ λπ α α− +
=−
= − − +−∑
where as it is usually :
(135) ( ) ( )( , ) [ , '( ), ]G G tm t If t Gt
αα α αα−
= =−
In the following we will apply the following considerations to the kernel (78) Factorization of the kernel (78) on the gudermann line in the w-plane The possible advantages to work in the w- plane (for instance by using the gudermann line) are two. The first is the possibility to facilitate function-theoretic manipulations involving integrals. To this end we remember that analytical continuations of approximate solutions are provided by using difference equations(section 1.4.4). To discuss the second advantage we observe that the samples of ˆ ( )F w+ are uniformly distributed in a w-line. Instead in the planeα − the samples of ( )F α+ are uniformly distributed in a α -line. The possible convenience is constituted by the fact that in the w-plane we experienced narrows spectral bands. For the numerical solution we can remain in the planeα − by considering the mapping of
(136) 1( ) ( )2 2
w y gd y j yπ= − + − − in the planeα − :
180
fig.6.40: the gundermann line in the planeα −
(137) 1cos ( ) cos[ ( ) ]2 2
k w y k gd y j yπα = − = − − + − −
Apparently the line gdα is very similar to the line / 4jke πλ = (fig. 6.28). However now we have very different samples. In the following we will obtain numerical solution of the Fredholm equation (84) by assuming A=3, h=0.1 (meromorfefredholm5_d2.nb).
Fig. 6.41 Comparison on the exact and approximate solutions on the gundermann line
181
Fig. 6.42 Error ( ) ( )( )( )
aF FeFα αα
α+ +
+
−=
We compared also the exact values of , 1 1( ), ( )b c aF Fα α− + − with the approximate values
1( )a aF α+ −
1( ) -0.300053-j0.0625363 bF α− =
1( ) -0.30607-j0.0648318 a bF α− =
1( ) -0.580089-j0.0530757 cF α− =
1( ) -0.585687-j0.0550864 a cF α− =
1( ) -0.0766216-j0.134265 aF α+ − =
1( ) -0.0844624-0.124685 ä a aF α+ − = 1.6.3 2.1 Presence of poles of the kernel in the shaded region of fig .6.28. For illustrative purposes, let us consider the presence of a pole sα in the shaded region of fig.6.28. This pole may be present in the plus function )(α+F (zero of the )](det[ 1 α−G ) or in the minus function )(α−F (zero of the )](det[ αG ). By following the same procedure used for deducing eq. (71) it is possible to obtain directly the following integral equation on the αλ line
a) if sα is a pole of )(α−F
(138) [ ]
,0]Im[,)()()(
.2
1)()( <−
+−
=−
−+ ∫ +
+ os
o
o
o XFdx
xxFGxG
jFG α
αααααα
παα
αλ
182
where oX is unknown. It must be noticed that the last unknown term is present only if 0]Im[ >sα
b) if sα is a pole of )(α+F
(139) [ ]
,0]Im[,)()()(
.2
1)()( <−
+−
=−
−+ ∫ +
+ os
o
o
o YFdx
xxFGxG
jFG α
αααααα
παα
αλ
where oY is unknown. It must be noticed that the last unknown term is present only if 0]Im[ <sα In order to evaluate oX it can be used the sampled form of the equation:
(140) ,0)(
.2
1=
−∫ +
αλ απdx
xxF
j s
Similarly, to evaluate oY we can use the sampled form of the equation:
(141) ,0)(
.2
1=
+∫ +
αλ απdx
xxF
j s
1.6.3.2.2 An example Let’s consider the solution of the W-H equation
0
11( ) ( ) ( )0
G F Fα α αα α+ −= +−
where the kernel is the rational matrix factorized in 1.5.4.2
2 2
2 21( ) ( ) ( )
1
nAB
jqG G G
jq
ααα α α+
+− += = ⋅
with ( ) and ( )G Gα α− + are given in 1.5.4.2 In the numerical solution based on the MATHEMATICA Computer codes Fredholm_rational2 _1.nb,- Fredholm_rational2 _3.nb. it has been assumed
1, 2, 0.5, 5, 0.1nA B q A h= = = = = 11 10o jα −= − The plots relevant to the approximate solution ( )aF α+ obtained by Fredholm equations by using the different paths:
183
/ 4
( )( )( ) arctan
j
x xx xex x j x
π
λ
λλ
=
== +
and the exact solution :
1 1 1( ) ( )( ) 1
2
p
p
G GF
α αα
α α
− −+ −
+ =−
are the same. Concerning the error 1 1
1
( ) ( )( )( )
aF FeFα αα
α+ +
+
−= relevant to the first
components we have:
The error relevant to the path ( ) arctanx x j xλ = + is greater and it here has not been considered
1.6.2.1 The Fredholm equations for factorized kernels The method of Fredholm equation considered in the previous section yields directly the plus solution of the W-H equation. In order to have a Fredholm equation for the plus factorized kernel, according to sec.1.2.9, let us introduce the solution of the homogeneous W-H equation: (149) )()()( ααα −+ = ii UUG where the plus function )(α+iU and the minus function )(α−iU are not Laplace transforms, since they generally are not vanishing for ∞→α . Finding n independent
184
solutions )},(),({ αα −+ ii UU },.....2,1{ ni = of (94) it provides the factorized matrices of )()()( ααα +−= GGG through the equations:
(150) ),(,),........(),()( 21 αααα −−−− = nUUUG , 1
21 ),(,),........(),()( −++++ = αααα nUUUG
In order to obtain the Fredholm equation, let us introduce now the plus Laplace Transforms:
(151) o
ii
UX
ααα
α−
= ++
)()(
where oα is an arbitrary point with negative imaginary part. By use of this substitution equation (94) becomes:
(152) o
ii
UXG
ααα
αα−
= −+
)()()(
where the unknowns )(α+iX and o
iUααα
−− )(
are vanishing ∞→α .
The integration on the 1γ line defined in 1.1.5 yields:
(153) ,))((
)(.
21)()(
.2
111∫∫ −−
=−
−+
γγ ααπαπdx
xxxU
jdx
xxXxG
j o
ii
In the second member the only singularity present in the half-plane 0]Im[ <x is ox α= , and consequently
(154)
n
oo
oi
o
i
C
CC
Udx
xxxU
j ..1)(
))(()(
.2
1 2
1
1 ααααα
ααπ γ −=
−=
−−−−∫
where n is the order of the matrix kernel )(αG and niCi ,..2,1, = are unknown
constants. By taking into account that o
ii
UX
ααα
α−
= ++
)()( is a plus function, the integration
on the line 2γ defined in 1.1.5 yields:
(155) ,0)(
.2
)(2
=−∫ +
γ απα dx
xxX
jG i
185
By some easy algebraic manipulations on the previous equations it follows the Fredholm
equation on o
ii
UX
ααα
α−
= ++
)()( :
(156) [ ]
0]Im[,)()()(
.2
1)()( >−
=−
−+ ∫
∞
∞−
++ o
o
iii
Rdx
xxXGxG
jXG α
αααα
παα
where
1
2 ( )...i i o
n
CC
R U
C
α−= = .
By forcing
(157) ,
1...00
.....,,
0...10
,
0...01
21 === nRRR
it is possible to obtain n independent solutions o
ii
UX
ααα
α−
= ++
)()( that through (95)
provide the factorized plus )(α+G . Equation (97) can be rewritten:
(158) ( ) ( ) ( )( ) ( ) i i o i oi
o o
U U UG X α α αα αα α α α
− − −+
−= +
− −
Consequently the W-H technique yields the solution:
(159) 1 1 ( )( ) ( ) ( ) i oi o
o
UX G G αα α αα α
− − −+ + −=
−
Of course this solution satisfies the Fredholm equation (101). We can proof it by substitution of this expression in eq.(101).
Taking into account that o
ii
UX
ααα
α−
= ++
)()( , eq. (101) yields:
(160) [ ]( ) ( ) ( )( ) ( ) . , Im[ ] 0
2io
i i o
G x G X xG U dx R
j xαα αα α α
π α∞ +
+ −∞
−−+ = <
−∫
186
Even though it can be show that the integral do not present the pole oα , the vectors
( )iU α+ depends on oα . In fact 1 1( ) ( ). ( ). ( )i o o i o i oU G R G Uα α α α− −+ −= = . In the following
we proof that by changing oα yields values of ( , )oG α α+ that differs by a pre-multiplicative constant matrix. Let’s introduce the factorized matrices , ( )G α− + , 1( , )G α α+ , 2( , )G α α+ where , ( )G α− + are arbitrary standard factorized matrix of ( )G α and 1( , )G α α+ and 2( , )G α α+ have been construct with the previous technique by assuming 1oα α= and 2oα α= respectively. The Wiener-Hopf solutions yields the following expressions for the vectors 1
iX + and 1iX +
that through equation (95) provide the expressions of 1( , )G α α+ and 2( , )G α α+ :
(161) 1 1
1,21,2
1,2
( ) ( )i i
G GX R
α αα α
− −+ −
+
⋅=
−
Algebraic manipulations that take into account equations (95) and (96) yield: (162) 2 1( , ) ( , )G K Gα α α α+ += where the constant matrix K is given by:
(163) 11 1 1 1
2 1 2 2 2 3 2 4
1 1 1 11 1 1 2 1 3 1 4
( ) , ( ) , ( ) , ( )
( ) , ( ) , ( ) , ( )
K G R G R G R G R
G R G R G R G R
α α α α
α α α α
−− − − −− − − −
− − − −− − − −
= ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅
For what concerns the location of the pole oα , note that if NullSpace[ ( )]o Gα α∈ , it follows ( ) 0i oU α− = . Similarly if 1NullSpace[ ( )]o Gα α−∈ , it follows ( ) 0i oU α+ = . By excluding these sets, apparently it seems that the only constraints is Im[ ] 0oα < . We did not study the problem of the better choice of oα . However we warning that a not suitable choice of oα could have an unsatisfactory numerical solution of the Fredholm equation (101). In particular we introduce with oα a new singularity and it increases the possibility to have significant errors in same region of the planeα − (see comments done in sect. about the correctness of the representation of plus functions trough sampled equation ( )). Consequently we suggest to define oα in the context of a physical problem related to the factorization of the matrix ( )G α . For instance in open structures a physical source is constituted by a plane wave and we can assume oα such that it is a pole related to this wave. 1.6.4.2.1 examples
187
In this example we will apply the consideration of 1.6.4.2 to factorize the rational matrix considered in the example of 1.5.4.2 .
2 2
2 21( )
1
nAB
jqG
jq
ααα+
+=
In the numerical solution based on the MATHEMATICA Computer code Fredholm_order2_fin.nb, it has been assumed
1, 2, 0.5, 5, 0.1nA B q A h= = = = = In order to obtain the factorized matrices as solutions of the homogeneous equations (149) the pole oα has been assumed
11 10o jα −= − The equations (150) provide the two factorized matrices ( )aG α− and ( )aG α+ that have been evaluated numerically in the MATHEMATICA Computer code Fredholm_rational2 _4.nb. The solution of the W-H equation:
11( ) ( ) ( ) 1
2p
G F Fα α αα α+ −= +−
with the source the pole 0.1p jα = − has been evaluated with the equation:
1 1 1( ) ( )( ) 1
2
a a pa
p
G GF
α αα
α α
− −+ −
+ =−
and compared with the exact solution :
1 1 1( ) ( )( ) 1
2
p
p
G GF
α αα
α α
− −+ −
+ =−
where the exact factorized matrices ( ) and ( )G Gα α+ − have been considered in 1.5.4.2
188
The errors 1,2 1,21,2
1,2
( ) ( )( )
( )aF F
eFα α
αα
+ +
+
−= of the two components of ( )aF α+ and ( )F α+ ,
in the range , are indicated in the fig. (Fredholm_rational_2_4.nb)
By repeating the same evaluation with the same 11 10o jα −= − , but by putting the physical pole very near the real axis:
81 10p jα −= − we obtained the errors indicated in fig. (Fredholm_rational_2_4.nb)
A=0.5 h=0.1 In order to compare with a direct solution of the W_H equation we considered the numerical direct solution of
189
11( ) ( ) ( )0o
G F Fα α αα α+ −= +−
By comparing with the direct solution considered in the example 1.6.3.2.2 it appears that using an arbitrary pole oα for obtaining the factorized matrices through the homogeneous solution is a valid tool. Example 3.1 Rational matrices of order two (Fredholm_rational2_4prova.nb) In this example we compare the exact and the approximate factorization of the rational matrix:
(70.1)
2 2
2 21( ) ( ) ( )1
ajqG G Gbjq
αα α αα − +
+ = = ⋅+
where exact factorized matrices ( ) and G ( )G α α− + are given by:
( )( )( )
2 2 2
2
2 2 2
2
2 2 2 2
2 2 2 2
1
1
1 1
1( )
0
q b a q
q
b a q
q
j b a q j q
b q b a q b j bqG
j
b
α
αα
+
+−
+
+
+ + +
− + + + + −=
−− +
( )( )( )
( )( )( )
2 2 2
22 2 2
2
2 2 2 2 2 2 2 2
1
1
1 10
( )q b a q
qq b a q
q
b q b a q b a q j q
q b jG
bq jbbq
q b j
α
αα
α
α
++
++
+
− + + + + − +
−=
− +− −
−
Fredholm approximate factorized matrices have been obtained with the method discussed previously by assuming 1, 2 and q=0.5a b= = . Structural poles are given by
1.844, 2j j± ± and are sufficiently far from the real axis . By assuming the spurious
190
source pole 1 0.1p jα = − , the real axis as integration line and the parameters 10 and h=0.1A = , we compared the solution of the W-H equations
1 1( ) ( ) ( )oo
RF G Gα α αα α
− −+ + −= ⋅ ⋅
−
where 1 20.5R R R= + , 1 2
1 0,
0 1R R= =
by using exact and approximate factorized matrices ( ) and G ( )G α α− + . In all the worked cases the plots of absolute values of exact and approximate components of ( )F α+ are identical. So we plotted relative errors. Fig. 8 illustrates the relative errors on the two components of ( )F α+ in the case of location of the source pole 0.1o jα = −
fig.8 relative errors for the two components of the plus functions in the case of physical
source pole located in 0.1o jα = − . 1
2
( )( )
( )F
FF
αα
α+
++
= means value obtained by exact
factorizations, 1
2
( )( )
( )a
aa
FF
Fα
αα
++
+
= means value obtained by approximate factorizations.
Fig. 9 illustrates the relative errors on the two components of ( )F α+ in the case of location of the source pole 0.1o jα = , i.e in a different half-plane of location of
1 0.1p jα = −
191
fig.9 relative errors for the two components of the plus functions in the case of physical source pole located in 0.1o jα = + . In evaluate the importance of relative error, take into account that the (identical) plots of of the exact and the approximate values of ( )F α+ are reported in fig.10
fig.10 Absolute values of the components of ( )F α+ . Case 0.1o jα = + Example 3.2 Approximate Fredholm factorization of matrices of order three. Fredholm_rational_ordine3a2be.nb Let’s consider the factorization of the rational matrix:
192
(70.2)
2
2 2
2
2
2
2 2
421 1
9( ) 1 21
1 21 1
j
G
αα α
ααα
αα α
+ + + +
= + + +
that has as inverse:
( )( )( )
( )( )( )
( )( )
( )( )( )
( ) ( )( )( )
( ) ( )( )
22 4 2 2 2 4 2
6 4 2 6 4 2 6 4 2
2 4 2 2 4 2 2 4 21
6 4 2 6 4 2 6
1 4 7 5 1 4 7 10 1
4 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5
1 5 2 1 4 2 2 1 5 4 9( )
4 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5 4
j j j j j j j j j
j j j jG
j j j j j j
α α α α α α α
α α α α α α α α α
α α α α α α α α αα
α α α α α α α−
+ + − + + +− −
+ − − + + + − − + + + − − + +
+ − + + + − − + + − + −= −
+ − − + + + − − + + + ( )
( )( )( )( )
( ) ( )( )
4 2
22 4 2 2 26 4
6 4 2 6 4 2 6 4 2
1 4 (3 7 ) 5
1 2 2 2 1 12 3 14 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5 4 1 4 (3 7 ) 5
j j j
j jj j j j j j j j j
α α
α α α α αα αα α α α α α α α α
− − + +
+ + − + + − + −− + − − + + + − − + + + − − + +
The exact factorization of this matrix was accomplished with the method indicated in the introduction by putting 2p jα = − . The expressions of the exact factorized matrices are very involved and are not reported here (matr_rat_grado_3bis.nb). In order to use the approximate factorization method described in this paper it is important to ascertain the location of the structural singularities i.e. the zeroes of det[ ( )]G α and 1det[ ( )]G α−
fig.11a: location of the structural singularities of the kernel ( )G α fig.11b: alternative integration lines in the Fredholm equation
193
The fig.11a reports the location of these poles. We observe that two of the zeroes (± (0.746591 +0.0409929j)) of det[ ( )]G α are very near the real axis. Both the components of ( )G α and 1( )G α− are bounded so that the only warning to consider if one wish use the real axis as integration line, is that of the location of the two zeroes of det[ ( )]G α near the real axis. To ascertain possible troubles we performed numerical
simulations with different lines integration defined by 3
4( ) ,j
y e y yπ
α = −∞ < < ∞ ,
2
arctan( )10
yy y jx
α = −+
, y−∞ < < ∞ and ( ) arctany y j yα = − , y−∞ < < ∞ (fig.11).
In the region between the real axis and these lines, no structural singularities are present. Since the simulations show that using the real axis the computer time is very small and we have an acceptable accuracy, we report only the approximate factorized solution by working with real axis. Fig.12 illustrates the comparison between the exact and approximate solution for the three components of the plus function ( )F α+ . We used the following parameters A=14, h=0.1 and 2o jα = − . Fig.13 reports the relative error
( ) ( )( ) , i=1,2,3( )
i i ai
i
F x F xe xF x
+ +
+
−=
fig.12 Comparison for the three components of the plus functions in the case of physical source pole located in 2o jα = − . The spurious peaks near 0.75 are due to the offending minus pole near the real axis in the approximate solution.
194
Fig.13 relative errors for the three components of the plus functions in the case of physical source pole located in 2o jα = − . Example : Diffraction by a PEC wedge at skew incidence. This problem has been considered in {IEEE). The involved matrix is of order 4 and can be factorized in closed form (SIAM). Approximate factorized matrices have been obtained with the technique indicated in the previous section. In the upper part, Fig. 4 illustrate the exact and approximate diffraction coefficient obtained for this problem (the two plots are indistinguishable). In the lower part is reported the relative error in 10log scale.
195
Fig. 4 Convergence of the approximate wedge diffraction coefficients to the exact ones by increasing the truncation parameter A and diminishing the discretation
parameter h. The relative error is reported in 10log scale. 1.6.3 Improving the approximate solution in the framework of Fredholm
equations Approximate solutions can be improved by rewriting the W-H equation in the form:
(181) o
oaaaaa
FGFGFGGGG
αααααααααα
−+= −
−−−−++
−+
−− )()()()()()()()( 1111
where )(α±aG are approximate factorized matrices of the kernel )(αG . It yields the new W-H equation:
(182) o
ooam
FGXXGαα
αααα
−⋅
+=−−
−+
)()()()(
1
where: (183) )()()( ααα +++ = FGX a
196
(184) )()()()( 11 αααα −+
−−= aam GGGG
It is possible to obtain new approximate solutions by
(185) o
ooaaa
FGGFGXαα
ααααα
−−
+=−−
−−
−−−−
))()(()()()(
111
Associated to this W-H equation, there is the Fredholm equation
(186) [ ]
0]Im[,)()()()(
.2
1)()(1
<−
⋅=
−−
+−−∞+
∞−
++ ∫ o
o
ooammm
FGdxx
xXGxGj
XG ααα
ααα
παα
The last equation constitutes a new starting point of the previously developed procedure. There are many possible chooses of )(α±aG , for instance, a suitable )(α+aG can match the asymptotic behavior of the solution (near field) as ∞→α . Furthermore, by taking into account that the formal W-H solution is expressed by:
(187) o
oo FGGFαα
ααα
−⋅
=−−−
++
)()()(
11
it is possible to approximate it in a such way that (188) )()( ooa GG αα −− ≈ and, similarly, for what concerns the factorized plus matrix when α represents the observation point. In general, considering (189) )()()( αεαα iiGG += with known factorized matrices )(α−iG and )(α+iG of )(αiG (190) )()()( ααα +− ⋅= iii GGG and )(αε i vanishing in a given point iα , it follows
(191) )()()()()()()()( 111 ααεααα
ααααα +−−
−−−
−−++ ⋅−
−⋅+⋅= FG
FGFGFG ii
o
oiii
By assuming the plus part, it yields:
197
(192) dtt
tFttGj
FGFG ii
o
ooii ∫ −
⋅−
−⋅+= +
−−−
−++1
)()()(2
1)()()(1
1
γ αε
πααααα
where 1γ is the path indicated in fig. 3.1 of section 1.3.1. Introducing iαα = , it yields:
(193) dtt
tFttGj
GFFi
iiiiiii ∫ −−= +
−−
−+
++ γ αε
πα
αα)()()(
2)(
)()(11
where
(194) o
ooiii
FGGF
ααααα
−⋅+= −
−−++ )()()( 11
By taking into account that 0)( =ii αε , in iαα = , the error is given by the term: (195)
∫
∫∫∞
∞−
+−−
−+
+
−∞
∞−
+−−
−++
−−
−+
−−=
=−−
−=−
−
i
iiii
iiiii
i
iiii
i
iiii
ttFttG
jG
FGt
tFttGj
Gdtt
tFttGj
G
αε
πα
ααεα
αε
πα
αε
πα
γ
)()()(2
)(
)()(2
)()()()(2
)()()()(2
)(
11
11111
1
Due to the presence of the factor it α−
1 in the integral, and since 0)( ≈αε i as iαα ≈ ,
this error is small, provided that )(~)()(~ 11 tGttG −+
−− ε is sufficiently vanishing as ∞→t .
1.6.4 Comments on the approximate methods for solving W-H equations In the previous sections, we described many approximate methods for solving W-H equations. In practice we experienced that the more powerful are those based on rational approximants of the kernel and on the quadrature method for solving Fredholm integral equations. These techniques provide approximate factorized kernels that are accurate in their regularity regions. This does not constitute a limitation since, outside these regions, we can make use of the equation (196) )()()( ααα +−= GGG For instance, approximate values of )(1 α±
+G in points where 0]Im[ <α , may be evaluated by: (197) 111 ))()(()( ±−
−±+ = ααα GGG
198
Consequently, since )(1 α±G is exactly known everywhere, the problem of evaluating the plus factorized matrix outside its regularity half-plane, is reduced to the evaluation of
)(1 α±−G in its regularity half-plane 0]Im[ <α .
The power of the previously introduced approximate methods depends on the spectrum of the kernel, e.g. if there is not an infinite discrete spectrum, together with a continuous spectrum, we suggest to work on the Fredholm equation in the w-plane that, in this case, provides very accurate approximate solutions. Conversely, if it is present only an infinite discrete spectrum, accurate solutions can be obtained by the rational approximant of kernels and, in particular, we experienced that the interpolation methods work better than Pade approximants. In all the cases the difficulties arise mainly from the presence of singularities near the real axis, and so it is always convenient to try to eliminate these singularities by using appropriate algebraic manipulations. The Fredholm equation relevant to modified kernel An important characteristic of the Fredholm factorization discussed in the previous sections is constituted by the fact that it is not necessary to regularize the operator having as symbol ( )G α by performing preliminary factorizations of scalar of matrices. To improve the accuracy of the factorization, some times it is convenient to generalize the Fredholm factorization by introducing a slight modification of the original technique discussed in…. Let rewrite ( )G α ( ) ( ) ( )m pG G Gα α α= where the inverse of ( )mG α is regular in the lower half plane Im[ ] 0α ≤ . By repeting the procedure that yields the Fredholm equation we obtain:
1( ) ( ) ( ) ( )1( ) ( ) .2
p p m op
o
G t G F t G RG F dtj t
α αα απ α α α
−+∞ ++ −∞
− ⋅ + =− −∫ Im[ ] 0oα <
We have obtained another Fredholm equation that sometimes can be more convenient of the original one. For instance if ( ) and ( )m pG Gα α are the standard factorized matrices of ( )G α
( ) ( ) ( )G G Gα α α− += the integral is vanishing and it yields the expected results:
199
1 1( ) ( )( ) o
o
G G RF α ααα α
− −+ −
+
⋅ ⋅=
−
In this case we say that ( ) ( )mG Gα α−= produce a total regularization of the matrix kernel ( )G α In general we can choose the matrix ( )mG α in order to have a partial regularization of
( )G α . This expedient could simplify the numerical solution of the integral equation.
Some particular techniques of Fredholm factorization. Sometimes the kernel ( )G α has a form that do not allows the reduction to Fredholm integral equations as indicated in the section 1.2.11.5 . For instance let’ consider the kernel:
(1) ( ) ( )
( )( ) ( )
j L
j L
A B eG
C e D
α
α
α αα
α α
+
−
=
associated to the W-H equation
(2) ( ) ( ) ( )p
RG F Xα α αα α+ += +−
, Im[ ] 0pα <
Firstly we observe that it is not restrictive to consider the parameter L not negative. If it were not so we reduce to the equation (1) observing that:
0 1 0 1( ) ( ) ( ) ( )1 0 1 0( ) ( ) ( ) ( )
j L j L
j L j L
A B e D C eC e D B e A
α α
α α
α α α αα α α α
− +
+ −
=
Particular kernels having the previous form have been considered in the section… where we discussed the factorization of kernel relevant to the problem of staggered half-planes . For instance we observed that the case of partial overlapped half-planes ( 0L ≤ ) can be solved by a relatively simple technique. The success of a relatively simple technique was made possible for the particular form of the matrices ( ), ( ), ( ) and ( )A B C Dα α α α . In general this technique does not apply and the difficulties for factorizing matrices having the form (1) were overcame in the paper Abrahams I.D and Wickham: General Wiener-Hopf factorization of matrix kernels with exponential phase factors, SIAM J. Appl.Math., 50 ,819-838, 1990 The reason of the encountered difficulties is essentially due to the presence of j Le α+ . In fact this factor does not allow to close the line 1γ with a semi circle located at ∞ in the lower half-plane α . It follows the impossibility to deduce straightforward the Fredholm equation relevant to the W-H matrix kernels. To eliminate this inconvenient we first rewrite the matrix ( )G α in the form:
(3)1 0 1 0( ) ( )
( ) . ( ).0 0( ) ( )
j L
j L j Lj L
A B eG M
e eC e D
α
α αα
α αα α
α α
+
−−
= =
where:
( ) ( )( )
( ) ( )A B
MC Dα α
αα α
=
Next we suppose that the central matrix ( )M α is factorized (analytically or numerically) in the form: (4) ( ) ( ) ( )M M Mα α α− += It yields: (5) ( ) ( ). ( )G G Gα α α− += ! ! where:
1 0( ) . ( )
0 j LG Me αα α− −−
=
! , 1 0
( ) ( ).0 j LG M
e αα α+ +
=
!
1 1 1 0( ) ( ) .
0 j LG Me αα α− −
− − +
=
! , 1 11 0
( ) ( )0 j LG M
e αα α− −+ +−
= ⋅
!
The factorized matrices ( ) and ( )G Gα α− +
! ! and their inverses are regular respectively in the lower and upper half-planes Im[ ] 0α ≤ and Im[ ] 0α ≥ . However they do not constitute standard factorized matrices since ( )G α−
! grows exponentially in the half-plane
Im[ ] 0α ≥ and ( )G α+! grows exponentially in the half-plane Im[ ] 0α ≤ .
To eliminate this inconvenient we rewrite eq. (2) in the form:
(6) 1 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )p pp p
R RG F G G X G G wα α α α α α α αα α α α
− − − −+ + − − + − − − = + − = − −! ! ! ! !
Since the first member is regular in the half-plane Im[ ] 0α ≥ and the second member is regular in the half-plane Im[ ] 0α ≤ , it follows that the function ( )w α is entire. Moreover it is vanishing in Im[ ] 0α ≥ and conversely grows exponentially in Im[ ] 0α ≤ . Multiplying eq.6 by j Le α− yields that ( )j Le wα α− it is vanishing in Im[ ] 0α ≤ and conversely grows exponentially in Im[ ] 0α ≥ . These properties of the entire function
( )w α will be utilized later. To deduce the Fredholm integral equation for ( )w α we use the result obtained from eq.
(6) 1 1
1 ( ) ( )( ) ( ) ( ) p
p
G G RF G w
α αα α α
α α
− −+ −−
+ +
⋅ ⋅= +
−
! !!
By integrating on the “frown” real axis 2γ we rewrite:
2 2 2
1 11 ( ) ( )( ) ( ) ( )1 1 12 2 2
o
p
G u GF u G u w u Rdu du duj u j u j u uγ γ γ
απ α π α π α α
− −−+ −+ + ⋅
= + ⋅− − − −∫ ∫ ∫
! !!
Now the first member is vanishing since we can close the line 2γ with a half-circle with radius →∞ located in the half-plane Im[ ] 0u ≥ ,where ( )F u+ is regular and vanishing as u →∞ . It follows:
2 2
1 11
0
( ) ( )( ) ( )1 12 2
oG u GG u w u Rdu duj u j u uγ γ
απ α π α α
− −−+ −+ ⋅
= − ⋅− − −∫ ∫
! !!
or:
(7) 2
1 111 ( ) ( )( ) ( )1 1 1( ) ( ) . .
2 2 2o
p
G u GG u w u RG w PV du duj u j u uγ
αα απ α π α α
− −−∞− + −++ −∞
⋅− + = − ⋅
− − −∫ ∫! !!!
Taking into account that ( )j Le wα α− is regular and vanishing in Im[ ] 0α ≤ . We have:
(8) 1
1 ( ) 02
juLe w u duj uγπ α
−
=−∫
In fact we can close the “smile” real axis with a half-circle with radius →∞ located in the half-plane Im[ ] 0u ≤ Equation (8) can be rewritten:
(9)( )
1
1 11( ) ( ) ( )( ) 1 1( ) ( ) . 0
2 2 2
j u Lj L juLG e G e w ue w u du G w PV duj u j u
αα
γ
α αα απ α π α
−− −− ∞−+ ++ −∞
= + =− −∫ ∫
! !!
Subtracting (7) from (8) yields:
( )
2
1 1 1 11
( ) ( ) ( ) ( ) ( )1 1( ) ( )2 2
j u L
o
p
G e G u w u G u G RG w du duj u j u u
α
γ
α αα απ α π α α
−− − − −∞ + +− + −+ −∞
− ⋅ + = ⋅− − −∫ ∫
! ! ! !!
or (10)
( )
2
1 1 1( ) ( ) ( ) ( ) ( ) ( )1 1( )2 2
j u L
o
p
e G G u w u G G u G Rw du duj u j u u
α
γ
α α ααπ α π α α
− − − −∞ + + + + −
−∞
− ⋅ + = ⋅− − −∫ ∫
1 ! ! ! ! !
For evaluating the second member we observe that for the presence of 1( )G u−
+! that grows
exponentially for Im[ ] 0u ≥ , we cannot close the line 2γ with an infinite half-circle located in the upper half-plane. Conversely we can close 2γ with an infinite half-circle located in the lower half-plane. Taking into account that this close line enclose the poles p and α α and all the singularities of 1( )G u−
+! , it yields:
2
2
1 1
1 1 1 1 1
( ) ( ) ( )12
( ) ( ) ( ) ( ) ( ) ( ) ( )12
o
p
p o o
p p
G G u G R duj u u
G G G G R G G u G R duj u u
γ
λ
α απ α α
α α α α α αα α π α α
− −+ + −
− − − − −+ + + − + + −
⋅⋅ =
− −
− ⋅ ⋅ = + ⋅− − −
∫
∫
! ! !
! ! ! ! ! ! !
where 2λ is a closed line that enclose only the singularities of 1( )G u−
+! .It is interesting to
observe that the known term of the integral equation does not present the source pole pα . Now we show that equation (10) reduces to a Fredholm equation of second kind when the matrix on the real axis 1( )M α−
+ reduce to a constant matrix oA as α → ±∞ . First taking into account that
1 11 0( ) ( )
0 j LG Me αα α− −
+ +−
= ⋅
!
we rewrite (10) in the form:
( ) ( )
2
1 1
1 1
0 0( ) ( ) ( )
0 1 0 11( ) ( )2
( ) ( ) ( )12
j L juLjuL
o
p
e eM M u e w u
w G duj u
G G u G R duj u u
α
γ
αα α
π α
α απ α α
− − −+ +
∞
+ −∞
− −+ + −
−
+ +−
⋅= ⋅
− −
∫
∫
!
! ! !
or
( ) ( )
2
1 1
1 1
0 0( ) ( ) ( )
0 1 0 11( ) ( )2
0 0( )
0 1 0 1 ( ) ( ) ( )1 1( )2 2
j L juLjuL
o o
j L juLjuL
oo
p
e eM A M u A e w u
w G duj u
e ee A w u
G G u G RG du duj u j u u
α
α
γ
αα α
π α
α ααπ α π α α
− − −+ +
∞
+ −∞
−
− −∞+ + −
+ −∞
− − −
+ +−
−
⋅ + = ⋅− − −
∫
∫ ∫
!
! ! !!
The last integral of the first member can be evaluated in closed form:
1 1
0 0( )
0 1 0 11 ( )2
0( )
0 10 ( )1 1( ) ( )2 20 1
j L juLjuL
o
juLjuL
oj L juLo
e ee A w u
G duj u
ee A w u
e e A w uG du G duj u j u
α
α
γ γ
απ α
α απ α π α
−
∞
+ −∞
−−
+ +
−
=−
= + − −
∫
∫ ∫
!
! !
where 1γ is the “smile” real axis. Taking into account the properties of ( )w α , in the first integral the “smile” real axis can be closed with an infinite half-circle located in the lower half-plane and it shows that this integral is vanishing. In the second integral the “smile” real axis can be closed with an infinite half-circle located in the upper half-plane and it yields:
1
0( )
1 00 11 ( ) ( ) ( ) ( ) ( )02
juLjuL
o
o oj L
ee A w u
G G A w M A wej u αγ
α α α α απ α
−
+ + +−
= = − ∫! !
Equation (10) can be rewritten in the form:
(11)
( ) ( )
2
1 1
1 1
0 0( ) ( ) ( )
0 1 0 111 ( ) ( ) ( )2
( ) ( ) ( )12
j L juLjuL
o o
o
o
p
e eM A M u A e w u
M A w G duj u
G G u G R duj u u
α
γ
αα α α
π α
α απ α α
− − −+ +
∞
+ + −∞
− −+ + −
− − −
+ + + −
⋅= ⋅
− −
∫
∫
!
! ! !
This equation is of the second kind since we can show the compactness of the operator
( ) ( )1 10 0( ) ( )
0 1 0 1
j L juL
o oe eM A M u A
u
α
α
α
− −+ +
− − −
−
by using the same considerations done in the sect. …. The presence of unknown ( )w α that grows exponentially in the lower half-plane do not allow to deform the real axis to improve the convergence of the numerical solution of this equation. However we can consider the characteristic of ( )w α for indicating a way to solve the integral equation alternative to the quadrature method (A-W). Let’ consider the inverse transforming ˆ ( )w x of ( )w α :
1ˆ ( ) ( )2
j xw x w e dj
αα απ
∞ −
−∞= ∫
For 0x < we can close the real axis with an infinite half-circle located in the upper half-plane. It yields: ˆ ( ) 0w x = , per 0x <
rewriting in the following form
( )1ˆ ( ) ( )2
j L j L xw x w e e dj
α αα απ
∞ − −
−∞= ∫
and taking the property of ( ) j Lw e αα − in the lower half-plane, for 0L x− < or x L> we can close with an infinite half-circle located in the lower half-plane. yielding: ˆ ( ) 0w x = , per x L>
Consequently the function ˆ ( )w x is not vanishing only in the interval 0 x L< < and its Fourier transform ( )w α can be represented by the sampling theorem. More directly we observe that the function:
( )1j L
we α
α−
is a meromorphic function vanishing at ∞ . The Mittag-Leffler expansion of this function yields the series (sampling theorem):
( )( )1 ( )
nj L
n n
wwe jLα
ααα α
∞
=−∞
=− −∑
where 2n n
Lπα =
It leads to the following representation of ( )w α through the sampling ( )nw α :
( )1 ( )
( )( )
j Ln
n n
e ww
jL
α αα
α α
∞
=−∞
−=
−∑
Substituting in the integral of the Fredholm equation (11) and forcing
mα α= ( 0, 1, 2,..)m = ± ± yields a system of infinite unknowns ( )nw α that according to the compactness of the kernel can be solving by truncation. A simplification of the system of the integral equations (11) can be accomplished by observing that choosing suitably ( ), ( ), ( ) and S ( ),P Q Rα α α α− − − − we can rewrite:
( ) ( ) ( ) 0 ( ) 1( ) .
( ) ( ) 0 ( ) ( )j L j L j L j L
p q P RG
r e s e Q S e eα α α α
α α α αα
α α α α− − − −
− − − − −− − − −
= =
!
and similarly:
1 ( ) 0 1 ( )( ) .
0 ( ) ( )j L j L
P RG
Q e S eα α
α αα
α α− + ++ − −
+ +
=
!
By introducing the plus unknown 1
( ) 0( )
0 ( )P
FQ
αα
α
−
++
+
and the minus unknown
1( ) 0
( )0 ( )
PX
Qα
αα
−
−+
−
it shows that it is not restrictive to suppose that the factorized
matrices ( )G α−! and 1( )G α−
+! present the form:
( ) 1( )
( ) j L j L
RG
S e eα α
αα
α−
− − −−
=
!
1 1 ( )( )
( )j L j L
RG
e S eα α
αα
α− ++ − −
+
=
!
For instance it yields:
1 1 ( )( )
1 ( )R
MS
αα
α− ++
+
=
and 1 ( )1 ( )o
RA
S+
+
∞=
∞
the integrand of (11) becomes ( agg_abra.nb)
( ) ( )1 1
2
0 0( ) ( ) ( )
0 1 0 1
0 ( ) ( ) ( ) ( )( )
0 ( ) ( )
( ) ( ) ( ) ( )( )
( ) ( )
j L juLjuL
o o
j L juLjuL
j L juLjuL
e eM A M u A e w u
e R R e R u Re w u
S S u
e R R e R u Re w u
S S u
α
α
α
α
α
α
α
α
− − −+ +
+ + + + −
+ +
+ + + + −
+ +
− − − =
− ∞ − − ∞ = = −
− ∞ − − ∞ =−
where 2 ( )w α is the second component of the vector 1
2
( )( )
( )w
ww
αα
α=
It means that the bottom component of the system of the two integral equations :
(12)
( ) ( )
2
1 1
1
11 1
0 0( ) ( ) ( )
0 1 0 11( ) 1 ( ) ( )2
1 ( ) ( ) ( ) ( )2
j L juLjuL
o o
o
o o
p
e eM A M u A e w u
w M A G duj u
M A G G u G R duj u u
α
γ
αα α α
π α
α α απ α α
− − −+ +
∞−
+ + −∞
−− −
+ + + −
− − −
+ + + −
+ ⋅ = ⋅− −
∫
∫
!
! ! !
do not present the unknown 1( )w α . It simplify the problem since reduces the vector problem to two uncoupled scalar equations Quralche problema longitudinale puo essere affrontato con questo metodo per esempio
1( )
( ) 0
j L
j L
eG
e Z
α
αα τ
α−
=
yields:
1
( )1 0S
M α ±±
=
, 1 0 1( )
1M
Sα−
±±
= −
0 11 0oA
=
Tuttavia e’ meglio usare il metodo di Jones Fare il metodo di Jones per longitudinali e trasversali. Approximate solution of longitudinal modified W-H equations
Approximate rational Factorization of the matrix 0
( )( )
j L
j L
eG
Z e
α
αα
α
− =
The factorization of the triangular matrix ( )G α is fundamental for solving the longitudinal modified W-H equations (sect 1.1.2.1) that arise from finite convolution equations. The factor j Le α is a plus function but its inverse j Le α− is a minus function so the factorization of this matrix introduces an interesting mathematical problem that have been considered by many authors. For instance a vast literature concerns with the case where ( )Z α is an almost periodical polynomial (see for example [64,65]). Besides for what concerns the solution of the W-H equation related to ( )G α in electromagnetic problems many ideas are reported in literature (see example 6.8, p.233 of Noble [3].
A possible way to overcome the difficulties related to the presence of j Le α and j Le α− in the kernel is to rationalize by pade approximants .For instance the Pade representation of
jxe with the function
an(x)f (x)=Pade[Exp[ I x],{x,0,N,N}]= d(x)
For instance, by assuming 20N = yields the relative error ( )( )jx
ajx
e f xe xe−
= illustrated
in fig:
In particular 16 9( 20) 1.68783 10 3.81168 10e j− −± = − ⋅ ± ⋅ It must be observed that the polynomials di degree N satisfies the equation:
( ) ( )n x d x= − .It means that ( ) ( )n x d x− is an even function in x. Besides the numerator ( ) ( )n x n x−= is a minus functions (i.e. has N zeroes with positive imaginary part) and the
denominator ( ) ( )d x d x+= is a plus functions (i.e. has N zeroes with negative imaginary part) Using a rational representation of j Le α allows to factorize the matrix ( )G α according the method described in 1.5.2. It yields:
1 0( )( )
( ) ( ) ( )n LG
n L S n Lαα
α α α−−
− − −
=
, 1
( ) 0( ) 1( ) ( )
( )
n LG
n L Sn L
αα
α αα
−−−
− −−
= −
( ) 0( ) 1( ) ( )
( )
d LG
d L Sd L
αα
α αα
+
++ +
+
=
, 1
1 0( )( )
( ) ( ) ( )d LG
d L S d Lαα
α α α
−++
+ + +
= −
where S± follows fro the additive decomposition of:
( ) ( ) ( )
( ) ( )Z S S
n L d Lα α α
α α − +− +
= +
The factorized matrices present a polynomial behavior as α →∞ so that by separating the plus from the minus functions the W-h technique introduces not vanishing polynomial functions. To see in detail these aspects let’s consider the W-H equations:
1 1
2 2
( ) ( ) 01( )( ) ( ) 1o
X XG
X Xα α
αα α α α
+ −
+ −
⋅ = +−
where Im[ ] 0oα <
After factorizing
( ) 0( )0
( )( ) ( )( )
j L
j L
d Ln Le
n LZ e Zd L
α
α
αα
αα αα
+−
−
−
+
≈
the separation in two members
plus and minus yields the equations:
11
11
( )( ) ,( )( )( )
( )
wXd LwX
n L
αααααα
++
−−
=
=
2 1 21( ) ( ) ( ) ( ) ( )
( )( )o o
X d L S w wn L
α α α α αα α α+ + +
−
= − + −
2 1 21 1( ) ( ) ( ) ( ) ( )
( )( ) ( )( )o o o
X n L S w wn L n L
α α α α αα α α α α α− − −
− −
= − + + + − −
where 1 2( ) and ( )w wα α are polynomials that must be chosen in order to have vanishing values of 1,2X ± as α →∞ . Firstly we observe from the expressions of 1X ± that the degree of 1( ) w α must be 1N − . Consequently 1X ± are vanishing α →∞ . Considering the expression of 2X + if follows also that the degree of 2 ( ) w α must be
2N − .
Taking into account that the dominant part of
1 21 1 ( ) ( ) ( )
( )( ) ( )( )o o o
S w wn L n L
α α αα α α α α α −
− −
− + + +− −
as α →∞ is
1 21 ( ) ( ) ( )
( )( )o o
S w wn L
α α αα α α +
−
− +−
We have:
1 2
2 3 12 3 0 1 1
1 ( ) ( ) ( )( )( )
1... ...
o o
N N NN N N N
S w wn L
BBA A A O
α α αα α α
α αα α α
+−
− −− − +
− + =−
= + + + + + + +
If we force the 1 2 1N N N− + = − conditions:
2 3 0.. 0N NA A A− −= = = =
1 0.. 0N NB B B−= = = = we can observe :
1 2 1
1 1( ) ( ) ( )( )( ) N
o o
S w w On L
α α αα α α α+ +
−
− + = − as α →∞ ,
it means that the functions 2X + is vanishing α →∞ . Moreover since the dominant part of
1 21 1 ( ) ( ) ( )
( )( ) ( )( )o o o
S w wn L n L
α α αα α α α α α −
− −
− + + +− −
as α →∞ is
1 21 ( ) ( ) ( )
( )( )o o
S w wn L
α α αα α α +
−
− +−
also the functions 2X − is vanishing α →∞ . Taking into account that the polynomial 1 2( ) and ( )w wα α have 1 2 1N N N+ − = − arbitrary coefficients the conditions we force constitute a linear system of 2 1N − equations in the unknowns coefficients of 1 2( ) and ( )w wα α . To evaluate the coefficients of the system let’s consider (106) of 1.5.10.2. It yields:
1
2 1
1 1 1...... [ ]( )( ) ( ) ( ) ( )
No o
N No o o o o
On L n L n L n L
α αα α α α α α α α α α
−
+− − − −
= + + + +−
2
2 2
2 1 2
2 112 1 2
1 1 1( ) ( ) ( )( ) ( ) ( ) ( )
1 1...... ( ) [ ]( ) ( )
1...... [ ]
N
N N
NN N
tS Z t dt Z t dtn t L d t L n t L d t L
tZ t dt On t L d t L
SS O
αα α
α α
α α α
∞ ∞
+ −∞ −∞− + − +
−∞
− −∞− +
−−
= − −
− + =
= + + +
∫ ∫
∫
where:
1
( ) 1, 2,.., 2 1( ) ( )
i
itS Z t dt i N
n t L d t L
−∞
−∞− +
= − = −∫ ,
Application to the strip problem. The matrix ( )G α involved in the strip problem where the source is an E-polarized plane wave with normal incidence is given by:
2 2
0( ) 1
j L
j L
eG
ek
α
αα
α
− = −
where L is the length of the strip and k is the propagation constant.
We assumed 81, 1 10 ,2okL k j α−= = − = and used the previous algorithm by
putting N=4, N=8, N=10 and N=20 (strip6..nb). Because of ( )Z α is an even function of α the coefficient iS are vanishing for i odd. Conversely by assuming an high value of N, the coefficients iS assume very high values for high values of i. In this occurrence there are difficulties for numerically solving the system. However as indicate in the figure 1 ( )X α+ is in practice the same for very different values of N.
( )11( )
( )( )0 0( )
j L
j L
n kke d kkGk d kek n k
α
α
α ααα ααα
α α αα α α
−
++ −− = ≈ = − − + +
=
1( )
0
j L
j L
kekG
kek
α
α
ααα
αα
−
+
− = −
+
Approximate solution with Fredholm equations (traditional method). In order to introduce a compact operator we rewrite the W-H equation in the form:
1 1
2 2
0 ( ) ( ) 0( ) ( ) 1
1
j L
o
oj L
ke F F kkF Fke
k
α
α
αα α ααα α α αα
α
−
+ −
+ −
− ++ ⋅ = + −+
−
where
1 11( ) ( )F X
kα α
α+ +=−
, 1 11( ) ( )F X
kα α
α− −=+
2 2( ) ( )F k Xα α α+ += − 2 2( ) ( ) o
o
k kF k X
α αα α α
α α− +
+ − += + +
−
It yields the Fredholm equation (eq.(49) of 1.6.2):
1
2
1
2
0 ( )( )
1
0
0 ( ) 01( ) 12
j L
j L
j L jtL
j L jtL
o
o
ke FkFke
k
k k te ek k t
k k te e F t kk k t dtF tj t
α
α
α
α
ααααα
α
αα
ααα
π α α α
−
+
+
− −
∞ +
−∞+
−
+ ⋅ + +
− − −
− + + + +
− +− − + ⋅ =− −∫
The presence of the two exponentials j Le α and j Le α− do not allow to use the expedients considered in 1.6.2.2.3 to improve the convergence of the numerical solution of this equation. So we attempt to solve it by using the real axis. To dealing with a compact operator we truncate with say B the interval of integration:
1
2
1
2
0 ( )( )
1
0
0 ( ) 01( ) 12
j L
j L
j L jtL
j L jtL
B o
Bo
ke FkFke
k
k k te ek k t
k k te e F t kk k t dtF tj t
α
α
α
α
ααααα
α
αα
ααα
π α α α
−
+
+
− −
+
−+
−
+ ⋅ + +
− − −
− + + + +
− +− − + ⋅ =− −∫
However numerical simulation are not satisfying so we suggest to use the method ideated by Jones to reduce the longitudinal modified W-H equations to Fredholm equations. The main advantage of this method is that we face with a Fredholm equation where only one of two exponentials j Le α or j Le α− is involved . By this way we can introduce a path where the only involved exponential (say j Le α ) is rapidly decreasing for increasing values of α . However with respect the classical Fredholm factorization considered in the previous section, this advantage is paid by the necessity to factorize the element ( )Z α . Approximate solution with Fredholm equations (Jones method)
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