Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS

which converges providedc>a>0.

Prove the result (18.150).

SinceF(a, b, c;x) is unchanged by swappingaandb, we may write its integral representation
(18.144) as

F(a, b, c;x)=

Γ(c)

Γ(a)Γ(c−a)

∫ 1

0

ta−^1 (1−t)c−a−^1 (1−tx)−bdt.

Settingx=z/band taking the limitb→∞,weobtain

M(a, c;z)=

Γ(c)

Γ(a)Γ(c−a)

∫ 1

0

ta−^1 (1−t)c−a−^1 lim

b→∞

(

1 −

tz

b

)−b

dt.

Since the limit is equal toetz, we obtain result (18.150).

Relationships between confluent hypergeometric functions

A large number of relationships exist between confluent hypergeometric functions

with different arguments. These are straightforwardly derived using the integral

representation (18.150) or the series form (18.148). Here, we simply note two

useful examples, which read

M(a, c;x)=exM(c−a, c;−x), (18.151)

M′(a, c;x)=

a

c

M(a+1,c+1;x), (18.152)

where the prime in the second relation denotesd/dx. The first result follows

straightforwardly from the integral representation, and the second result may be

proved from the series expansion (see exercise 18.19).

In an analogous manner to that used for the ordinary hypergeometric func-

tions, one may also derive relationships betweenM(a, c;x) and any two of the

four ‘contiguous functions’M(a± 1 ,c;x)andM(a, c±1;x). These serve as the

recurrence relations for the confluent hypergeometric functions. An example of

such a relationship is

(c−a)M(a− 1 ,c;x)+(2a−c+x)M(a, c;x)−aM(a+1,c;x)=0.

18.12 The gamma function and related functions

Many times in this chapter, and often throughout the rest of the book, we have

made mention of the gamma function and related functions such as the beta and

error functions. Although not derived as the solutions of important second-order

ODEs, these convenient functions appear in a number of contexts, and so here

we gather together some of their properties. This final section should be regarded

merely as a reference containing some useful relations obeyed by these functions;

a minimum of formal proofs is given.