13 confluent hypergeometric surfaces

7/28/2019 13 Confluent Hypergeometric Surfaces

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13 Confluent HypergeometricFunctionsL U C YO A N L A T E B '

ContentsMathematical Properties . . . . . . . . . . . . . . . . . . .13.1. Definitionsof Kummer and Whittaker Functions . . . . .13.2. Integral Representations . . . . . . . . . . . . . . .13.3. Connections With Bessel Functions . . . . . . . . . . .13.4. Rkcurrence Relations and DifTerential Properties . . . . .13.5. Asymptotic Expansionsand Limiting Forms . . . . . . .13.6. Special C ases . . . . . . . . . . . . . . . . . . . .13.7. Zerosand TurningValues . . . . . . . . . . . . . . .

Page504504505506506508509510Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 511

13.8. Use and Extension of theTables . . . . . . . . . . . . 51113.9. Calculation of Zeros and TurningPoints . . . . . . . . . 51313.10. GraphingM(a. b z) . . . . . . . . . . . . . . . . . . 513References . . . . . . . . . . . . . . . . . . . . . . . . . . 514Table 13.1. Confluent Hypergeometric FunctionM(a. b. z) . . . . . 516Table 13.2. Zerosof M(u. b. z) . . . . . . . . . . . . . . . . . . 535

Z= . l .1) l(1) 10.a=- (.1)l. b= .1(.l)l, 8s~=-1(.1) -.1, b=.I( . l) I , 7D

The tables were calculated by the author on the electronic calculator EDSACI in theMathematical Laboratory of Cambridge University. by kind permission of its director. Dr .hl V Wilkes. The table of M(a. b. 2) was recomputed by Alfred E Beam for uniformityto eight significant figures

* University Mathematical Laboratory. Cambridge (Prepared under contract with theNatiOd Bureau O StaIldlUdS.)
http://266d8ad0e2e2ab9aeb5a5cd59bb972.pdf/http://92501cb4f9debaeae9c5a1f2558205.pdf/http://92501cb4f9debaeae9c5a1f2558205.pdf/http://266d8ad0e2e2ab9aeb5a5cd59bb972.pdf/


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13. Confluent Hypergeometric FunctionsMathematical Properties

13.1. D efinitions of K ummer and W hittakerFunctionsK ummer's Equation

?!!+(b-z) dw-uw=odZ dz3.1.1I t hasaregularsingularity at z=O and an irregularsingularity at m .Independent solutions are

K ummer's Function13.1.2

where( u) ~= u( u+ ~) ( u+ ~). . (u+n-1), (u) ~=I ,

and13.1.3

P arameters(m, n positive integers) M(a, b, 4all values of a, b and zi n 2

b# -n a#-m a convergent series forb# -n a=-m a polynomial of degree mb=-n a#-mb=-n a=-m, a simple pole at b=-n

m>nb=-n a=-m, undefinedU(a, b, z) isdefined even when b+fn13.1.4

m5nAs 121+mJ

M(a,b, z)=m'z"-b[l+O(lzl-l)] (9z>O)r (a)and13.1.5

U(a, b, z) isa many-valued function. Itsprinci-pal branch isgiven by -


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CONFLUENT HYPERGEOMETR IC FUNCTIONS 50513.1.15 y4=z1-bezM(1--a,2-b, - 2)

13.1.17 ~s=~'-~U( l+a-b,-b, Z)13.1.18 y,=e'U(b--a, b, - 2)13.1.19 y,=zl-bezU(l--a, 2-4 - 2)

WmIM kianSIf W {m, n}=y,y~-y,,y& andt=sgn ( J z)=l if .fz>o,=-1 if Y Z l O13.1.20

W{1, 2}=W{3,4}=W{l,4}=-W{2, 3)=(14 ) z - bez13.1.21W {1,3}=W {2,4}=W{5,}=W{7,8}=0

13.1.22 W {1, 5)=-r (b)~-~e~/ r(a)13.1.23 W {1, 7)=r(b)e"~*~-~e~/ r(b--a)13J .24 W{2, 5) =- (2- )z- "*/r(+a- 3)13.1.25 W{2, 7}=- (2- )z- bez/ r(-a)13.1.26 W (5, 7}=e"f(b-a' e*

K ummer T ransformationsM (a, b, z)=e'M(b--a, b, - 2)3.1.27

13.1.28~-~M (l+a-b, -4 z)=zl-*ezM(l-u, 2-b, - 2)13.1.29 U(a, b, 2)=z1-*U(l+a-6, 216, Z)13.1.30

Whittaker's Equation13.1.31 %+I-,+--+ 1 K (t-r')],=,z

Solutions:W hittaker's Functions

13.1.32 Mg, , (z)=e-+z++M3+p-N 1+2p,Z)13.1.33Wg.,(Z)= e-W + W (+p--K, 1+2p, 2)

(-r


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506 CONFLUENT HYPERGEOMETRIC F mycmoN 813.2.8 r (a)~( a ,, Z)

=eAzJAm e-z,(,-A )a-'(t+B )b-a-l~t(A=l-B)

Smlar integrals for ME,&)and W #,,,(z) canbe deduced with the help of ..13.1.32'and 13.1.33.13.2.9 Barnes- typeContour Integrals

for larg (-z)l


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CONFLUENT HYPERGEOMETRIC FUNCTIONS 50713.4.6(a-l+z)M(a, b, z)+(b-a)M(a-l, b, 2)

+(1-b)M(a, b-1, z)=O13.4.7b(l-b+z)M(a, b, z)+b(b-l)M(a-1, b-1, Pi

-azM(a+l, b + l , z)=O

(a)"M(a+n, b+n, 2)"dz"3.4.9 - M (a, b, Z) }13.4.10 aM(a+l, b, z)=aM(~ , z)+zM '(a, b, Z)13.4.11(b-a)M(a-1, b, z)=(b-a-z)M(a, b, 2)

+zM '(a, b, 413.4.12

(b-a)M(a, b+ l , z)=bM (a, b, z)-bM '(a, b, 2)13.4.13(b-l)M(a, 6-1, z)=(b-l)M(a, b, 2)+zM'(a, bJ z13.4.14(b-l)M(u-l, 6-1, z)=(b-1-z)M(a, b, 2)

+zM'(aJ J z,13.4.15U(a-1, b, z)+(b-2a-z)U(a, b, 2)

+a(l+a-b)U(a+l, b, z)=O13.4.16(b--a-l)U(fZ, b-1, z)+(l-b-z)U(a, b, 2)

+zU(a, b+l, z)=O13.4.11

U(a, b, 2)-aU(a+l, b, 2)-U(a, b-1, z)=O13.4.18(b-a) U(a, b, 2) +U(a- 1, b, 2)-zU(a, b+l, z)=O

13.4.19(a+z)U(a, b, z)-zU(a, b+l, z)

+a@-a-l)U(a+l, b, 2)=013.4.20(a+z-l)U(a, b, 2)-U(a-1, b, z)

+(l+a-b)U(a, b-1, z)=O13.4.21 U'(U,b, z)=-aU(a+l, b + l , Z)13.4.22

13.4.23a(l+a-b)U(a+l, b, z)=aU(a, b, 2)

+zU'@, b, 2)13.4.24(l+a-b)U(a, b-1, z)=(l-b)U(a, b, 2)

-zU'(a, b, 2)13.4.25 U(a, b + l J z)=U(U,b, z)-U'(U, b, Z)13.4.26U(a-1, b, z)=(a-b+z)U(a, b, z)-zU'(a, b, 2)

13.4.27U(u-1, b-1, z)=(l--b+z)U(a, b 2)-zU'(a, b, 2)


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08 C 0 " T HYPEBGEOMETI UC FUNCI'IONS

13.5.1113.5.12

(b=O)

~L S 4 - m for b bounded, z real.

where u ie defined in 13.5.13.

aaa+-- for b bounded, x rad.For largereal a,b xIf cdah' 6=~/ ( 2b-4~) 80 that ~>2b-U >l ,


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CONFLUENT HYPERGEOMETRIC F " C M 0 N B 509If z= (2b-&)[l+t/ (b--2a)~], so that

Z-2b-4~13.5.19M(a, b, z)=e+=(b-2a)'-Or(b)[Ai(t) cos (UT)13.5.20U(a, b, z)=e+=+"-+T(+)-+&-*

+B i (t) sin (UT)+O(I4b-a I-*)]

i--tr (~)(bz--2az)-13f~-f+O( l~--al-i)}

13.5.21M(a,b,z)=r(b)exp (b-24 COS* e}

[(b-2~) COSe]'-'[~$b-u) sinm]-+[sin (ad+sin (+-a) (2e-sin 2e)+ir)

13.5.22U(U,b, ~) = exp(b-24 C O S~B ][(~-~U ) COS el1-*[(3b-U) sin2e)-*{sin ($&a)

If cos*f?=z/(2b-4~ o that 2b--4a>z>O, I (20- sin 26)+tT I+O(l3b--al-') 113.6. SpeCi.1Casea

13.6.113.6.213.6.313.6.413.6.513.6.613.6.713.6.813.6.913.6.1013.6.1113.6.1213.6.1313.6.1413.6.1513.6.1613.6.1713.6.1813.6.1913.6.20

Relation

e*

*

Function

B eeSel&See1ModifiedBesselSpherical BesaelSpherical BesaelSpherical BesaelKelvinCoulomb Wave

Incomplete GammaPoisson-CharlierExponentialTrigonometricHyperbolic

WeberorParabolic CylinderHermiteHermiteError IntegralToronto

*See page 11.


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CONFLUENT HYPERGEOMETR IC F U " I [ O N S 511The self-adjoint equation 13.1.1 can ala0 bewritten

13.7.6

Ihe Sonine-P olya T heoremThe maxima and minima of Iwl form an in-creasing or decreasing sequence according as

Ie - ' e -& Numerica13.8. Use and Extension of theTables

C alculation of M (a, b, x)K ummer's T ransformation

Compute M(.3, .2, -.I ) to 7s.Using 13.1.27 and Tables4.4 and 13.1 we havea=& b=.2 so thatM (.3, .2,-1) =e-.'M(- l, .2, .l)Thus 13.127 can be used to extend Table13.1 tonegative values of z. Kummer's transformationshould also be used when a and b are large andnearly equal, for z largeor small.Example2. Compute M(17, 16, 1) to 7s.Here a=17, b=16, and

Exmple1.

=.85784 90.

M(17, 16, l)=elM(-l, 16, -1)=2.71828 18X1.0625000=2.88817 44.

R ecurrmaR elationsExample 3. Compute M(--1.3, 1.2, .l ) to 7s.Using 13.4.1 and Table 13.1 we have a=-.3,b=.2so that

M (- .3, .2, .1)=2[.7M (-3, .2, .1)- 3 M(.7, .2,. )]=.35821 23.

By 13.4.5 whena=- .3 and b= .2,M(-1.3,1.2, .1)=[.26 M(--3, .2, .l )-.24 M(--1.3, .2, .1)]/ .15=A9241 08.

Similarly when a=-3 and b= .2M(-.3, 1.2, .1)=.97459 52.

Check, by 13.4.6,M(-1.3, 1.2, .1)=[.2 M(-.3, .2, .l )

4-1.2 M(-.3, 1.2, .1)]/1.5=A9241 08.

isan increasingor decreasing function of z, that is,they form an increasing sequence for M (a, b, z)if a>O, zO and z>b-3 or if a< and

The turning values of Iw lie near the curvesz


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512 CONFLUENT HYPERGEOMETRIC FUNCTI ONSExample 6. Calculate M(.9, .l, 10) to 7S,using 13.5.1.

=-.198(.869) +1237253(.99190 285)=1227235.23- .17+O(1)=1227235+0(1)

+O(1)

Check, from Table13.1, M(.9, .l, 10)=1227235.To evaluateM(a, b, z) with a large, z small and bsmall or large 13.5.13-14 should beused.Example7. Compute M(-52.5, .l, 1) to 3s,usi ng 13.5.14.

M (-52.5, .l, 1)=r( l)e-'(.05+52.5).25-.M.5642COS [(.2-4( -52.5)) . I - 05r+ .25411+0((.05+52.5)-a6)]= -16.34+0(.2)

By direct application of a recurrence relation,M(-52.5, .l, 1) has been calculatedas -16.447.To evaluate M(a, b, z) with z, a and/or b large,13.5.17,19 or 21should be tried.Compute M(-52.5, .1, 1) usingxample8.

13.5.21 to 3s, COS e=42M(-52.5, .l ,1)-r(.l)e*oJ .loa2e1105.1COS 8J 1-.*.564152.55-1 sin 28-1[& (-52.5~)

+sin (52.55(2e-~n 2e)+tr)+O((52.55)-')]=- 6.47-t O(.02)A full range of asymptoticformulas to cover allpossible casesisnot yet known.

Calculationof U(a, b, x)For -105~510, -105~510, -105b510thisispossible by 13.1.3, usingTable13.1 and therecurrence relations 13.4.15-20.Example 9. Compute U(l.1, .2, 1) to 5s.Using Tables13.1, 4.12 and 6.1 and 13.1.3, wehave

U(.1, .2, 1)=

But M(.9, 1.8, 1)=.8[M(.9, . 8 ,)-M(-.l, .8, l)]= 1.72329,using 13.4.4.

HenceU(.1, .2, 1)=5.344799(.37 1765- 194486)

= 94752.SimilarlyHence by 13.4.15

U(-.9, .2, 1)=.91272.

U(l.1, 2, l)=[U(.l, .2, l)-U(-.9, .2, 1)]/ .09= 38664.Example10. To compute U'(-.9, - . 8 , 1) to5s. By 13.4.21

U'(-.9, -.8, 1)=.9U(.1, .2, 1)= (.9) (.94752)= 85276.AsymptoticFormulae

Example11. Tocompute U(1, .l ,100) to 5s.By 13.5.21 19 1929100 100100(1, .l ,100)=i&j{l-:+:

=.01{1-.019+.000551-.000021=.00981 53. +0(10-9) 1,

Example12. To evaluate V(.l, .2, .01). Forz small, 13.5.612 should be used.+O( .01)1--71- 2)r (1.1 -.2)(.l, .2, Ol)=

=-+O( (.01) 7U.9)=1.09 to 3S, by 13.5.10.

To evaluate U(u, b, z) with a large, z small andb smallor large 13.5.15or 16should beused.To evaluate V(a, b, z) with z, a and/or b large13.5.18, 20 or 22 should be tried. I n all thesecases the sizeof the remainder termisthe guidetothe numberof significant figures obtainable.

Calculationof theW hittaker FunctionaExample13. ComputeM .o.-.4(l)ndW.o, .4(1)to 5s. By formulas 13.1.32 and 13.1.33 andTables13.1, 4.4

-44.0, -.,(l) =e-,'M(.l, 2, 1)=1.10622,W.o.- ,(1) =e-.(U( 1, .2, 1)=.57469.


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Thus the values of M..,(z) and W d z ) canthe values of M (a, b, z) andb, z) are known.Calculation of Zeros and TurningPoints

a m p u b the smallest POsitiveUsing 13.7.2 we have, as a first

Ex-Ple 14.of M (-4, . 6 , ~>.This is outside the range ofable 13.2.

1;=x;l - M ( -3,.6,X i )-3M (-3, .6,X i)=X i [I -M (-2,1.6, X i)/ .6M (-3, .6,X i)]=.9715)0.M (--1.8, -.2, z) has a maximum in zwhen= .94291 9.

Compute the smallest positiveof x for which M (-3, .6, z) has a turningX i. T his is outside the range of Table13.2.13.4.8 we have

M (-1.8, -.2, z)=9M (-.8, .8, 2)

Example 16.

M (-3, .6, ~) = -3M ( -2, 1.6, ~) / .6.13.7.2 for M (-2, 1.6, z),

Xo=(1.0!k)2/ (11.2)= 9715.hisisafirst approximationto X iforM (-3, .6,z).sing 13.7.5 and 13.4.8 we find a second approxi-

FI GURE13.1.F igure 13.1 shows the curvesonwhich M (a,6, z)=O in the a, bplane when z=1. T he function ispositive in the unshaded areas, and negativein theshaded areas. T he number in each square givesthe number of real positive zeros of &&, b, z) as afunction of z in that square. T he verticalboundaries to the left are to be included in eachsquare. 13.10. Graphing M(a, b,x)Example 17. Sketch M (-4.5, 1, z). Firstly,

from F igure 13.1 we see that the function hasfive real positive zeros. From 13.5.1, we findthat M +- m , M + - m as x++ m and thatM ++m, M ++m as z+- - . B y 13.7.2 wehave6sfirstapproximations to the zeros, .3,1.5,3.7,6.9, 10.6, and by 13.7.2 and 13.4.8 we find as firstapproximations to the turning values .9, 2.8, 5.8,9.9. From 13.7.7, we see that these must lie nearthe curveay=f eN (54-t (1 -dl l )%-+.

From these facts we can form a rough graph ofthe behavior of the function, Figure 13.2.


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514 CONFLUENT HYPERGEOMETRIC F"C l ' I0NS

FIQUF~E3.2. M (-4.5, 1, 2).(From F. Gb2d?'ri,R~m$d;~y&~*&~o~l*dblonl.

FIQUBE3.4. M (a, .5, 2).Inc, New York, fi.Y ., 1945, with pmb l o n . )(Ffom E. J 8hnke cmd F. Emde T able8 of hmctlons Dover Publlcatknu,

ReferencesTcxts

[ 3.11 H. Buchholz, Die konfluente hypergeometrischeFunktion (Springer-V erlag, Berlin, Germany,1953). On W hittaker functions, with a largebibliography.(13.21 A. Erdelyi et al., H igher transcendental functions,vol. 1, ch. 6 (M cGraw-Hill Book Co., Inc., NewY ork, N.Y., 1953). On K ummer functions.[13.3] H. J effreys and B. 5. J effreys, M ethods of mathe-matical physics, ch. 23 (Cambridge Univ. P-,Cambridge; England, 1950). On K ummerfunctions.[13.4] J . C. P. M iller, Noteon the general solutionsof theconfluenthypergeometric equation, M ath. T ableaAidsComp.9,97-99 (1957).113.61 L. J . Slater, On the evaluation of the confluenthypergeometr ic function, P roc. C ambridgePhiloe. Soc. 49, 612-622 (1953).FIQWE 3.3. M(o, 1, z).ha., New York, &.Y, lM 6, withpemmbsbm.)(promE. J ahnkeand F Emde Tablesof function& Dover P ubllatkxu.


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CONFLUENT HYPERQEOMETBIC F U N C N O N S 515[13.6] L. J . Slater, The evaluation of the basic confluenthypergeometric function, Proc. CambridgePhilos. Soc. 50, 404-413 (1954).[13.7] L. J . Slater, Thereal mros of the confluent hyper-geometric function, Proc. Cambridge Philos.Soc. 52, 626-635 (1956).[13.8] C. A. Swanson and A. Erdhlyi, Asymptotic formsof confluent hypergeometric functions, Memoir

25, Amer. Math. Soc. 1957).[13.9] F. G. Tricomi, Funeioni ipergeometriche confluenti(Edizioni Cremonese, Rome, Italy, 1954). OnKummer functions.[13.10] E. T. Whittaker and G. N. Watson, A course ofmodern analysis, ch. 16, 4tb ed. (CembridgeUniv. Press, Cambridge, England, 1952). OnWhittaker functions.

T d h[13.11] J . R.Airey,The confluent hypergeometric unction,British Association Reports, Oxford, 276-294

(1926), and Lee& 220-244 (1927). M(a, b z),(.5)8, 5D.~=-4(.5)4, a=*, 1, 3, 2, 3, 4, ~=.1(.1)2(.2)3

(13.121 J . R. Airey and H. A. Webb, The practical impor-tance of the confluent hypergeometric function,Phil. Mag. 36, 129-141 (1918). M(a, b, z),(13.131 E. J ahnke andF. Emde, Tablesof functions, ch. 10,4th ed. (Dover Publications, Inc., New York,N.Y ., 1945). Graphs of M(a, b, z) based on thetables of [13.11].[13.14] P. Nath, Confluent hypergeometric functions,Sankhya J . Indian Statist. Soc. 11, 153-166(1951). M(u, b, z), a=1(1)40, b=3,2=.02(.02).1(.1)1(1)10(10)50, 100, 200, 6D.[13.15] 8. Rushton and E. D. Lang, Tables of the confluenthypergeometric function, Sankhye , Indian Statist.

Soc. 3, 369-411 (1954). M(a, b, Z), a=.5(.5)40,b= .5( 5)3.5, Z= .02 (.02).1(.1) 1 (1) 10(10) 50, 100[13.16] L. J . Slater, Confluent hypergeometric functions(Cambridge Univ. Preas, Cambridge, England,

~=-3(.5)4, b=1(1)7, z=1(1)6(2)10, 45.

200, 7s.

1960). M(u, b, z), ~=-l ( .l ) l , b=.l(.l)l,~=.l( .l) lO, 8s; M(u, b, l) , ~=-11(.2)2,b= -4(.2) 1, 85; and smallest positive values ofz for which Mfa, b, z)=O, a=-4(.1)-.l,b=.1(.1)2.5, 8s.

13 confluent hypergeometric surfaces

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