Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

second solution to (18.147) as a linear combination of (18.148) and (18.149) given

by

U(a, c;x)≡

π

sinπc

[

M(a, c;x)

Γ(a−c+1)Γ(c)

−x^1 −c

M(a−c+1, 2 −c;x)

Γ(a)Γ(2−c)

]

.

This has a well behaved limit ascapproaches an integer.

18.11.1 Properties of confluent hypergeometric functions

The properties of confluent hypergeometric functions can be derived from those

of ordinary hypergeometric functions by lettingx→x/band taking the limit

b→∞, in the same way as both the equation and its solution were derived. A

general procedure of this sort is called aconfluenceprocess.

Special cases

The general nature of the confluent hypergeometric equation allows one to write

a large number of elementary functions in terms of the confluent hypergeometric

functionsM(a, c;x). Once again, such identifications can be made from the series

expansion (18.148) directly, or by transformation of the confluent hypergeometric

equation into a more familiar equation for which the solutions are already

known. Some particular examples of well known special cases of the confluent

hypergeometric function are as follows:

M(a, a;x)=ex, M(1,2; 2x)=

exsinhx

x

,

M(−n,1;x)=Ln(x), M(−n, m+1;x)=

n!m!

(n+m)!

Lmn(x),

M(−n,^12 ;x^2 )=

(−1)nn!

(2n)!

H 2 n(x), M(−n,^32 ;x^2 ),=

(−1)nn!

2(2n+1)!

H 2 n+1(x)

x

,

M(ν+^12 , 2 ν+1;2ix)=ν!eix(

x

2

)−νJν(x), M(^12 ,^32 ;−x^2 )=

√

π

2 x

erf(x),

wherenandmare integers,Lmn(x) is an associated Legendre polynomial,Hn(x)is

a Hermite polynomial,Jν(x) is a Bessel function and erf(x) is the error function

discussed in section 18.12.4.

Integral representation

Using the integral representation (18.144) of the ordinary hypergeometric func-

tion, exchangingaandband carrying out the process of confluence gives

M(a, c, x)=

Γ(c)

Γ(a)Γ(c−a)

∫ 1

0

etxta−^1 (1−t)c−a−^1 dt,

(18.150)