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Appendix A
Sturm-Liouville, Legendre and Bessel Equations
A.1 Sturm-Liouville Equations
A.1.1 Regular equations
The eigenfunctions of a large class of ODEs form complete orthonormal systems in suit-able Hilbert spaces. To this class belong equations of the form
� �p .x/ u0�0 C q .x/ u D �w .x/ u .x/ ; a < x < b (A.1)
under the assumption that p; p0; q; and w are continuous and positive on Œa; b�. In sucha case (A.1) is called a regular Sturm-Liouville equation. We associate to (A.1) theboundary conditions:
˛u .a/ � ˇp .a/ u0 .a/ D 0
u .b/ � ıp .b/ u0 .b/ D 0;(A.2)
where the coefficients ˛; ˇ; ; ı are real numbers. To avoid trivial situations we shall as-sume the following normalization condition:
˛2 C ˇ2 D 2 C ı2 D 1:
In general, problem (A.1), (A.2) has nontrivial solutions only for special values of the pa-rameter �, called eigenvalues. The corresponding solutions are called eigenfunctions, andthey form the eigenspace associated to �. Let us introduce the (Hilbert) space L2w .a; b/of weighted square-integrable functions u on .a; b/with respect to the weight functionw:
L2w .a; b/ D´u W
Z b
a
u2 .x/w .x/ dx < 1μ
.
Then the following theorem holds:
© Springer International Publishing Switzerland 2015S. Salsa, G. Verzini, Partial Differential Equations in Action. Complements and Exercises,UNITEXT – La Matematica per il 3+2 87, DOI 10.1007/978-3-319-15416-9_A
406 Appendix A Sturm-Liouville, Legendre and Bessel Equations
Theorem A.1. There exists an increasing sequence of positive numbers ¹�j ºj�1 such that�j ! C1 and:
a) Problem (A.1), (A.2) admits a non-trivial solution if and only if � equals one of the �j .
b) For every j , �j is simple, that is the associated eigenspace has dimension 1.
c) The eigenfunction system ¹'j ºj�1 (suitably normalised) is an orthonormal basis inL2w .a; b/.
A.1.2 Legendre’s equation
When the coefficient p is zero, e.g. at a or b, the equation is irregular and the studybecomes more complicated. A classical case is that of Legendre’s equation��
1 � x2�u0�0 C �u D 0 � 1 < x < 1: (A.3)
In the applications this is coupled with boundary conditions of the type
u .�1/ finite, u .1/ finite. (A.4)
Particular solutions to (A.3), (A.4) are Legendre’s polynomials, defined by the Rodriguesformula:
Ln .x/ D 1
2nnŠ
dn
dxn
�x2 � 1�n .n � 0/:
Each polynomial Ln corresponds to the eigenvalue �n
D n .nC 1/. The first four Legen-dre polynomials are
L0 .x/ D 1, L1 .x/ D x, L2 .x/ D 1
2
�3x2 � 1� , L4 .x/ D 1
2
�5x3 � 3x� :
The following theorem holds:
Theorem A.2. In relationship to problem (A.3), (A.4):
a) There exists a non-zero solution if and only if
�D�n D n .nC 1/ ; n D 0; 1; 2; : : : :
b) For every n � 0 the solution corrisponding to �n is unique up to a constant factor, andcoincides with Legendre’s polynomial Ln.
c) The normalised polynomials ´r2nC 1
2Ln
μn�0
form an orthonormal system in L2 .�1; 1/.
A.2 Bessel’s Equation and Functions 407
Theorem A.2 allows to expand any f 2 L2 .�1; 1/ in Fourier-Legendre series:
f .x/ D1XnD0
fnLn .x/ ; where fn D 2nC 1
2
Z 1
�1f .x/Ln .x/ ; dx
with L2 .�1; 1/-convergence. We also have a result about pointwise convergence, in per-fect analogy with Fourier series.
Theorem A.3. If f and f 0 have at most a finite number of jump points in .�1; 1/, then
1XnD0
fnLn .x/ D f .xC/C f .x�/2
for every x 2 .�1; 1/.
A.2 Bessel’s Equation and Functions
A.2.1 Bessel functions
Here is a short summary of the main properties of Bessel functions. First, though, we needto introduce a function interpolating the values of the factorial nŠ: The gamma function� D � .z/ is
� .z/ DZ 1
0
e�t tz�1dt (A.5)
for z complex with Re z > 0. The function � is analytic for Re z > 0 and satisfies thefollowing relationships:
� .z C 1/ D z� .z/
� .z/ � .1 � z/ D �
sin�z.z ¤ 0; 1; 2; : : :/:
In particular,� .nC 1/ D nŠ .n D 0; 1; 2; : : :/
and
�
�nC 1
2
D 1 � 3 � 5 � � � � � 2n � 1
2n
p� .n D 1; 2; : : :/ :
One can define � .z/ for z real, negative and not integer, using
� .z/ D � .z C 1/
z:
In fact, we know how to compute � on .0; 1/, and the formula allows to find � on .�1; 0/.In general, once we know� on .�n;�nC 1/, we can compute it on .�n � 1;�n/. Finally,
408 Appendix A Sturm-Liouville, Legendre and Bessel Equations
�4 �3 �2 �1 1 2 3 4
�5
5
Fig. A.1 Graph of the gamma function on the real axis
coherently with (A.5) we define
� .�2n/ D �1 and � .�2n � 1/ D C1:
In this way � is defined on the entire real axis (Fig. A.1).
Bessel’s function of the first kind and order p, p real, is
Jp .z/ D1XkD0
.�1/k� .k C 1/ � .k C p C 1/
�z2
�pC2k.
In particular, if p D n � 0 is an integer (Fig. A.2):
Jn .z/ D1XkD0
.�1/kkŠ .k C n/Š
�z2
�nC2k:
When p D �n is a negative integer, the first n terms of the series vanish and
J�n .z/ D .�1/n Jn .z/ :
Hence Jn .z/ and J�n .z/ are linearly dependent.If p is not integer, for z ! 0 we have asymptotic behaviours:
Jp .z/ D 1
� .1C p/
�z2
�pCO �zpC2� , J�p .z/ D 1
� .1 � p/�z2
��pCO �z�pC2�so Jp .z/ and J�p .z/ are linearly independent.
A.2 Bessel’s Equation and Functions 409
5 10 15 20
�0:5
0:5
1
Fig. A.2 Graphs of J0 (solid), J1 (dashed) and J2 (dotted)
Functions of the first kind satisfy a number of identities:
d
dz
�zpJp .z/
� D zpJp�1 .z/ ,d
dz
�z�pJp .z/
� D �z�pJpC1 .z/ : (A.6)
In particularJ 00 .z/ D �J1 .z/ :
From these we also infer that for p D n C 12
(and only in that case), the correspondingBessel functions are elementary. For instance,
J 12.z/ D
r2
�zsin z, J� 1
2.z/ D
r2
�zcos z.
Particularly important are the zeroes of Jp . For any p, there is an infinite increasing se-quence
®˛pj j̄�1 of positive numbers such that
Jp�˛pj
� D 0 .j D 1; 2; : : :/:
When p is not an integer, every linear combination
c2Jp .z/C c2J�p .z/
is a Bessel function of the second kind. The (standard) function of the second kind is
Yp .z/ D cosp�Jp .z/ � J�p .z/sinp�
.
When p D n is integer, one defines1 (Fig. A.3)
Yn .z/ WD limp!nYp .z/
Note that Yp .z/ ! �1 when z ! 0C.
1 One can prove that the limit exists.
410 Appendix A Sturm-Liouville, Legendre and Bessel Equations
5 10 15 20
�1
�0:5
0:5
Fig. A.3 Graphs of Y0 (solid), Y1 (dashed) and Y2 (dotted)
A.2.2 Bessel’s equation
The Bessel functions Jp ,Yp are solutions of the so-called Bessel equation of order p � 0:
z2y00 C zy0 C �z2 � p2�y D 0:
The general integral is given, for any p � 0, by
y .z/ D c1Jp .z/C c2Yp .z/ .
In the most important applications, one is typically led to solve the (parametric) equation(with parameter �)
z2y00 C zy0 C��2z2 � p2
�y D 0 (A.7)
on a bounded interval .0; a/, with boundary conditions of the sort
y .0/ finite, y .a/ D 0. (A.8)
For these, the following theorem holds.
Theorem A.4. Problem (A.7), (A.8) has nontrivial solutions if and only if
� D �pj D�˛pja
�2.
In that case the solutions are
ypj .z/ D Jp
�˛pjaz�
up to multiplicative constants. Moreover, the normalised functionsp2
aJpC1�˛pj
�ypj
A.2 Bessel’s Equation and Functions 411
form an orthonormal basis in (w.z/ D z)
L2w .0; a/ D²u W kuk22;w D
Z a
0
u2 .z/ zdz < 1³
by virtue of the orthogonality relations:
2
a2J 2pC1�˛pj
� Z a
0
zJp��pj z
�Jp��pkz
�dz D
´0 j ¤ k
1 j D k:
With Theorem A.4 we can expand any f 2 L2w .0; a/ in Fourier-Bessel series:
f .z/ D1XjD1
fjJp��pj z
�; where fj D 2
a2J 2pC1�˛pj
� Z a
0
zf .z/ Jp��pj z
�dz;
with L2w .0; a/-convergence.Let us compute, for example, the expansion of f .x/ D 1 on the interval .0; 1/, with
p D 0:
fj D 2
J 21�˛pj
� Z 1
0
zJ0�˛0j z
�dz:
Using (A.6),d
dzŒzJ1 .z/� D zJ0 .z/
so we may write Z 1
0
zJ0�˛0j z
�dz D
1
�0jzJ1
�˛0j z
��10
D J1�˛0j
��0j
:
Finally
1 D1XjD1
2
�0jJ1�˛0j
�J0 �˛0j z�with convergence in norm L2w .0; 1/.
Also in this case one can insure pointwise convergence.
Theorem A.5. If f and f 0 have at most finitely many jump discontinuities on .0; a/, then
1XjD1
fjJp��pj z
� D f .zC/C f .z�/2
at every point z 2 .0; a/.
Appendix B
Identities
Here is a compilation of significant formulas and identities of common use.
B.1 Gradient, Divergence, Curl, Laplacian
Let F;u; v be vector fields and f; ' scalar fields, all assumed regular on R3.
Orthogonal Cartesian coordinates
1. gradient:
rf D @f
@xi C @f
@yj C @f
@zk
2. divergence:
div F D @
@xFx C @
@yFy C @
@zFz
3. Laplacian:
�f D @2f
@x2C @2f
@y2C @2f
@z2
4. curl:
curl F Dˇ̌̌̌ˇ̌ i j k@x @y @zFx Fy Fz
ˇ̌̌̌ˇ̌
Cylindrical coordinates
x D r cos �; y D r sin �; z D z .r > 0; 0 � � � 2�/
er D cos � iC sin � j, e� D � sin � iC cos � j; ez D k:
© Springer International Publishing Switzerland 2015S. Salsa, G. Verzini, Partial Differential Equations in Action. Complements and Exercises,UNITEXT – La Matematica per il 3+2 87, DOI 10.1007/978-3-319-15416-9_B
414 Appendix B Identities
1. gradient:
rf D @f
@rer C 1
r
@f
@�e� C @f
@zez
2. divergence:
div F D1
r
@
@r.rFr /C 1
r
@
@�F� C @
@zFz
3. Laplacian:
�f D @2f
@r2C 1
r
@f
@rC 1
r2@2f
@�2C @2f
@z2D 1
r
@
@r
�r@f
@r
C 1
r2@2f
@�2C @2f
@z2
4. curl:
curl F D1
r
ˇ̌̌̌ˇ̌ er re� ez@r @� @zFr rF� Fz
ˇ̌̌̌ˇ̌
Spherical coordinates
x D r cos � sin ; y D r sin � sin ; z D r cos .r > 0, 0 � � � 2� , 0 � � �/
er D cos � sin iC sin � sin jC cos k
e� D � sin � iC cos � j
ez D cos � cos iC sin � cos j� sin k:
1. gradient:
rf D @f
@rer C 1
r sin
@f
@�e� C 1
r
@f
@ e
2. divergence:
div F D @
@rFr C 2
rFr„ ƒ‚ …
radial part
C 1
r
1
sin
@
@�F� C @
@ F C cot F
�„ ƒ‚ …
spherical part
3. Laplacian:
�f D @2f
@r2C 2
r
@f
@r„ ƒ‚ …radial part
C 1
r2
²1
.sin /2@2f
@�2C @2f
@ 2C cot
@f
@
³„ ƒ‚ …
spherical part (Laplace-Beltrami operator)
4. curl:
curl F D 1
r2 sin
ˇ̌̌̌ˇ̌ er re r sin e�@r @ @�Fr rF r sin Fz
ˇ̌̌̌ˇ̌ :
B.2 Formulas 415
B.2 Formulas
Gauss’s formulas
The following formulas hold on Rn, n � 2, and we denote by:
• � a bounded domain with regular boundary @� and outward normal �.
• u; v vector fields that are regular1 up to the boundary of �.
• '; regular scalar fields up to the boundary of �.
• d� the infinitesimal surface element of @�:
We have the following formulas:
1.R div u dx D R
@ u � � d� (divergence formula)
2.R r' dx D R
@ '� d�
3.R �' dx D R
@ r' � � d� D R@ @�' d�
4.R div F dx D R
@ F � � d� � R
r � F dx
5.R �' dx D R
@ @�' d� � R r' � r dx (integration by parts)
6.R . �' � '� / dx D R
@ . @�' �'@� / d�
7.R curl u dx D � R@ u � � d�
8.R u � curl v dx D R
- v� curl u dxC R@ .v � u/ � � d�
Identities
div curl u D 0
curl grad' D 0
div .'u/ D ' div uCr' � u
curl .'u/ D ' curl uCr' � u
curl.u � v/ D .v � r/u� .u � r/ vC .div v/ u� .div u/ v
div.u � v/ D curl u � v � curl v � u
r .u � v/ D u � curl v C v � curl u C .u � r/ vC .v � r/u
.u � r/u D curl u � uC 12r juj2
curl curl u D r.div u/ ��u .curl curl D grad div � Laplacian/:
1 C 1���
is enough.
416 Appendix B Identities
B.3 Fourier Transforms
bu .�/ DZ
Ru .x/ e�i�x dx
General formulas
u buu .x � a/ e�ia�bu .�/eiaxu .x/ bu .� � a/
u .ax/ , a > 01
abu� �
a
u0 .x/ i�bu .�/xu .x/ ibu0 .�/.u � v/ .x/ bu .�/bv .�/u .x/ v .x/ .bu �bv/ .�/
Special transforms
u bue�ajxj, a > 0
2a
a2 C �2
1
a2 C x2�
ae�aj�j
e�ax2, a > 0
r�
ae� �2
4a
sin x
xe�jxj arctan
2
�2
�Œ�a;a� .x/ 2sin a�
�ı .x/ 1
1 2�ı .�/
B.4 Laplace Transforms 417
B.4 Laplace Transforms
eu .s/ DZ C1
0
u .t/ e�st dt
General formulas (u.t/ D 0 for t < 0)
u euu .t � a/ , a > 0 e�aseu .s/eatu .t/ , a 2 C eu .s � a/u .at/ , a > 0
1
aeu � s
a
�u0 .t/ seu .s/ � u.0C/u00 .t/ s2eu .s/ � u0.0C/ � su.0C/tu .t/ �eu0 .s/u .t/
t
R C1s
eu .�/ d�R t0u .�/ d�
eu .s/s
.u � v/ .t/ eu .s/ev .s/Special transforms
u euH .t/eat , a 2 C
1
s � aH .t/ sin at , a 2 R
a
s2 C a2
H .t/ cos at , a 2 Rs
s2 C a2
H .t/ sinh at , a 2 Ra
s2 � a2H .t/ cosh at , a 2 R
s
s2 � a2H .t/tn, n 2 N
nŠ
snC1
H .t/t˛, Re˛ > �1 �.˛ C 1/
s˛C1H .t/e�t2 es
2=4R C1s=2
e��2d�
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As of 2004, the books published in the series have been given a volume number.Titles in grey indicate editions out of print.As of 2011, the series also publishes books in English.
A. Bernasconi, B. CodenottiIntroduzione alla complessità computazionale1998, X+260 pp, ISBN 88-470-0020-3
A. Bernasconi, B. Codenotti, G. RestaMetodi matematici in complessità computazionale1999, X+364 pp, ISBN 88-470-0060-2
E. Salinelli, F. TomarelliModelli dinamici discreti2002, XII+354 pp, ISBN 88-470-0187-0
S. BoschAlgebra2003, VIII+380 pp, ISBN 88-470-0221-4
S. Graffi, M. Degli EspostiFisica matematica discreta2003, X+248 pp, ISBN 88-470-0212-5
S. Margarita, E. SalinelliMultiMath – Matematica Multimediale per l’Università2004, XX+270 pp, ISBN 88-470-0228-1
A. Quarteroni, R. Sacco, F.SaleriMatematica numerica (2a Ed.)2000, XIV+448 pp, ISBN 88-470-0077-72002, 2004 ristampa riveduta e corretta(1a edizione 1998, ISBN 88-470-0010-6)
13. A. Quarteroni, F. SaleriIntroduzione al Calcolo Scientifico (2a Ed.)2004, X+262 pp, ISBN 88-470-0256-7(1a edizione 2002, ISBN 88-470-0149-8)
14. S. SalsaEquazioni a derivate parziali - Metodi, modelli e applicazioni2004, XII+426 pp, ISBN 88-470-0259-1
15. G. RiccardiCalcolo differenziale ed integrale2004, XII+314 pp, ISBN 88-470-0285-0
16. M. ImpedovoMatematica generale con il calcolatore2005, X+526 pp, ISBN 88-470-0258-3
17. L. Formaggia, F. Saleri, A. VenezianiApplicazioni ed esercizi di modellistica numericaper problemi differenziali2005, VIII+396 pp, ISBN 88-470-0257-5
18. S. Salsa, G. VerziniEquazioni a derivate parziali – Complementi ed esercizi2005, VIII+406 pp, ISBN 88-470-0260-52007, ristampa con modifiche
19. C. Canuto, A. TabaccoAnalisi Matematica I (2a Ed.)2005, XII+448 pp, ISBN 88-470-0337-7(1a edizione, 2003, XII+376 pp, ISBN 88-470-0220-6)
20. F. Biagini, M. CampaninoElementi di Probabilità e Statistica2006, XII+236 pp, ISBN 88-470-0330-X
21. S. Leonesi, C. ToffaloriNumeri e Crittografia2006, VIII+178 pp, ISBN 88-470-0331-8
22. A. Quarteroni, F. SaleriIntroduzione al Calcolo Scientifico (3a Ed.)2006, X+306 pp, ISBN 88-470-0480-2
23. S. Leonesi, C. ToffaloriUn invito all’Algebra2006, XVII+432 pp, ISBN 88-470-0313-X
24. W.M. Baldoni, C. Ciliberto, G.M. Piacentini CattaneoAritmetica, Crittografia e Codici2006, XVI+518 pp, ISBN 88-470-0455-1
25. A. QuarteroniModellistica numerica per problemi differenziali (3a Ed.)2006, XIV+452 pp, ISBN 88-470-0493-4(1a edizione 2000, ISBN 88-470-0108-0)(2a edizione 2003, ISBN 88-470-0203-6)
26. M. Abate, F. TovenaCurve e superfici2006, XIV+394 pp, ISBN 88-470-0535-3
27. L. GiuzziCodici correttori2006, XVI+402 pp, ISBN 88-470-0539-6
28. L. RobbianoAlgebra lineare2007, XVI+210 pp, ISBN 88-470-0446-2
29. E. Rosazza Gianin, C. SgarraEsercizi di finanza matematica2007, X+184 pp, ISBN 978-88-470-0610-2
30. A. MachìGruppi – Una introduzione a idee e metodi della Teoria dei Gruppi2007, XII+350 pp, ISBN 978-88-470-0622-52010, ristampa con modifiche
31 Y. Biollay, A. Chaabouni, J. StubbeMatematica si parte!A cura di A. Quarteroni2007, XII+196 pp, ISBN 978-88-470-0675-1
32. M. ManettiTopologia2008, XII+298 pp, ISBN 978-88-470-0756-7
33. A. PascucciCalcolo stocastico per la finanza2008, XVI+518 pp, ISBN 978-88-470-0600-3
34. A. Quarteroni, R. Sacco, F. SaleriMatematica numerica (3a Ed.)2008, XVI+510 pp, ISBN 978-88-470-0782-6
35. P. Cannarsa, T. D’AprileIntroduzione alla teoria della misura e all’analisi funzionale2008, XII+268 pp, ISBN 978-88-470-0701-7
36. A. Quarteroni, F. SaleriCalcolo scientifico (4a Ed.)2008, XIV+358 pp, ISBN 978-88-470-0837-3
37. C. Canuto, A. TabaccoAnalisi Matematica I (3a Ed.)2008, XIV+452 pp, ISBN 978-88-470-0871-3
38. S. GabelliTeoria delle Equazioni e Teoria di Galois2008, XVI+410 pp, ISBN 978-88-470-0618-8
39. A. QuarteroniModellistica numerica per problemi differenziali (4a Ed.)2008, XVI+560 pp, ISBN 978-88-470-0841-0
40. C. Canuto, A. TabaccoAnalisi Matematica II2008, XVI+536 pp, ISBN 978-88-470-0873-12010, ristampa con modifiche
41. E. Salinelli, F. TomarelliModelli Dinamici Discreti (2a Ed.)2009, XIV+382 pp, ISBN 978-88-470-1075-8
42. S. Salsa, F.M.G. Vegni, A. Zaretti, P. ZuninoInvito alle equazioni a derivate parziali2009, XIV+440 pp, ISBN 978-88-470-1179-3
43. S. Dulli, S. Furini, E. PeronData mining2009, XIV+178 pp, ISBN 978-88-470-1162-5
44. A. Pascucci, W.J. RunggaldierFinanza Matematica2009, X+264 pp, ISBN 978-88-470-1441-1
45. S. SalsaEquazioni a derivate parziali – Metodi, modelli e applicazioni (2a Ed.)2010, XVI+614 pp, ISBN 978-88-470-1645-3
46. C. D’Angelo, A. QuarteroniMatematica Numerica – Esercizi, Laboratori e Progetti2010, VIII+374 pp, ISBN 978-88-470-1639-2
47. V. MorettiTeoria Spettrale e Meccanica Quantistica – Operatori in spazi di Hilbert2010, XVI+704 pp, ISBN 978-88-470-1610-1
48. C. Parenti, A. ParmeggianiAlgebra lineare ed equazioni differenziali ordinarie2010, VIII+208 pp, ISBN 978-88-470-1787-0
49. B. Korte, J. VygenOttimizzazione Combinatoria. Teoria e Algoritmi2010, XVI+662 pp, ISBN 978-88-470-1522-7
50. D. MundiciLogica: Metodo Breve2011, XII+126 pp, ISBN 978-88-470-1883-9
51. E. Fortuna, R. Frigerio, R. PardiniGeometria proiettiva. Problemi risolti e richiami di teoria2011, VIII+274 pp, ISBN 978-88-470-1746-7
52. C. PresillaElementi di Analisi Complessa. Funzioni di una variabile2011, XII+324 pp, ISBN 978-88-470-1829-7
53. L. Grippo, M. SciandroneMetodi di ottimizzazione non vincolata2011, XIV+614 pp, ISBN 978-88-470-1793-1
54. M. Abate, F. TovenaGeometria Differenziale2011, XIV+466 pp, ISBN 978-88-470-1919-5
55. M. Abate, F. TovenaCurves and Surfaces2011, XIV+390 pp, ISBN 978-88-470-1940-9
56. A. AmbrosettiAppunti sulle equazioni differenziali ordinarie2011, X+114 pp, ISBN 978-88-470-2393-2
57. L. Formaggia, F. Saleri, A. VenezianiSolving Numerical PDEs: Problems, Applications, Exercises2011, X+434 pp, ISBN 978-88-470-2411-3
58. A. MachìGroups. An Introduction to Ideas and Methods of the Theory of Groups2011, XIV+372 pp, ISBN 978-88-470-2420-5
59. A. Pascucci, W.J. RunggaldierFinancial Mathematics. Theory and Problems for Multi-period Models2011, X+288 pp, ISBN 978-88-470-2537-0
60. D. MundiciLogic: a Brief Course2012, XII+124 pp, ISBN 978-88-470-2360-4
61. A. MachìAlgebra for Symbolic Computation2012, VIII+174 pp, ISBN 978-88-470-2396-3
62. A. Quarteroni, F. Saleri, P. GervasioCalcolo Scientifico (5a ed.)2012, XVIII+450 pp, ISBN 978-88-470-2744-2
63. A. QuarteroniModellistica Numerica per Problemi Differenziali (5a ed.)2012, XVIII+628 pp, ISBN 978-88-470-2747-3
64. V. MorettiSpectral Theory and QuantumMechanicsWith an Introduction to the Algebraic Formulation2013, XVI+728 pp, ISBN 978-88-470-2834-0
65. S. Salsa, F.M.G. Vegni, A. Zaretti, P. ZuninoA Primer on PDEs. Models, Methods, Simulations2013, XIV+482 pp, ISBN 978-88-470-2861-6
66. V.I. ArnoldReal Algebraic Geometry2013, X+110 pp, ISBN 978-3-642–36242-2
67. F. Caravenna, P. Dai PraProbabilità. Un’introduzione attraverso modelli e applicazioni2013, X+396 pp, ISBN 978-88-470-2594-3
68. A. de Luca, F. D’AlessandroTeoria degli Automi Finiti2013, XII+316 pp, ISBN 978-88-470-5473-8
69. P. Biscari, T. Ruggeri, G. Saccomandi, M. VianelloMeccanica Razionale2013, XII+352 pp, ISBN 978-88-470-5696-3
70. E. Rosazza Gianin, C. SgarraMathematical Finance: Theory Review and Exercises. From BinomialModel to Risk Measures2013, X+278pp, ISBN 978-3-319-01356-5
71. E. Salinelli, F. TomarelliModelli Dinamici Discreti (3a Ed.)2014, XVI+394pp, ISBN 978-88-470-5503-2
72. C. PresillaElementi di Analisi Complessa. Funzioni di una variabile (2a Ed.)2014, XII+360pp, ISBN 978-88-470-5500-1
73. S. Ahmad, A. AmbrosettiA Textbook on Ordinary Differential Equations2014, XIV+324pp, ISBN 978-3-319-02128-7
74. A. Bermúdez, D. Gómez, P. SalgadoMathematical Models and Numerical Simulation in Electromagnetism2014, XVIII+430pp, ISBN 978-3-319-02948-1
75. A. QuarteroniMatematica Numerica. Esercizi, Laboratori e Progetti (2a Ed.)2013, XVIII+406pp, ISBN 978-88-470-5540-7
76. E. Salinelli, F. TomarelliDiscrete Dynamical Models2014, XVI+386pp, ISBN 978-3-319-02290-1
77. A. Quarteroni, R. Sacco, F. Saleri, P. GervasioMatematica Numerica (4a Ed.)2014, XVIII+532pp, ISBN 978-88-470-5643-5
78. M. ManettiTopologia (2a Ed.)2014, XII+334pp, ISBN 978-88-470-5661-9
79. M. Iannelli, A. PuglieseAn Introduction to Mathematical Population Dynamics.Along the trail of Volterra and Lotka2014, XIV+338pp, ISBN 978-3-319-03025-8
80. V.M. Abrusci, L. Tortora de FalcoLogica. Volume 12014, X+180pp, ISBN 978-88-470-5537-7
81. P. Biscari, T. Ruggeri, G. Saccomandi, M. VianelloMeccanica Razionale (2a Ed.)2014, XII+390pp, ISBN 978-88-470-5725-8
82. C. Canuto, A. TabaccoAnalisi Matematica I (4a Ed.)2014, XIV+508pp, ISBN 978-88-470-5722-7
83. C. Canuto, A. TabaccoAnalisi Matematica II (2a Ed.)2014, XII+576pp, ISBN 978-88-470-5728-9
84. C. Canuto, A. TabaccoMathematical Analysis I (2nd Ed.)2015, XIV+484pp, ISBN 978-3-319-12771-2
85. C. Canuto, A. TabaccoMathematical Analysis II (2nd Ed.)2015, XII+550pp, ISBN 978-3-319-12756-9
86. S. SalsaPartial Differential Equations in Action. FromModelling to Theory (2nd Ed.)2015, XVIII+688, ISBN 978-3-319-15092-5
87. S. Salsa, G. VerziniPartial Differential Equations in Action. Complements and Exercises2015, VIII+422, ISBN 978-3-319-15415-2
The online version of the books published in this series is available atSpringerLink.For further information, please visit the following link:http://www.springer.com/series/5418