Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS

Changing variables to plane polar coordinates (ρ, φ)givenbyx=ρcosφ,y=ρsinφ,we
obtain

Γ(n)Γ(m)=4

∫π/ 2

0

∫∞

0

ρ2(m+n−1)e−ρ

2

sin^2 m−^1 φcos^2 n−^1 φ ρ dρ dφ

=4

∫π/ 2

0

sin^2 m−^1 φcos^2 n−^1 φdφ

∫∞

0

ρ2(m+n)−^1 e−ρ

2

dρ

=B(m, n)Γ(m+n),

where in the last line we have used the results (18.157) and (18.164).

The result (18.165) is useful in proving the identity (18.158) satisfied by the

gamma function, since

Γ(n)Γ(1−n)=B(1−n, n)=

∫∞

0

yn−^1 dy

1+y

,

where, in the second equality, we have used the integral representation (18.163).

For 0<n<1 this integral can be evaluated using contour integration and has

the valueπ/(sinnπ) (see exercise 24.19), thereby proving result (18.158) for this

range ofn. Extensions to other ranges require more sophisticated methods.

18.12.3 The incomplete gamma function

In the definition (18.153) of the gamma function, we may divide the range of

integration into two parts and write

Γ(n)=

∫x

0

un−^1 e−udu+

∫∞

x

un−^1 e−udu≡γ(n, x)+Γ(n, x),

(18.166)

whereby we have defined theincomplete gamma functions γ(n, x)andΓ(n, x),

respectively. The choice of which of these two functions to use is merely a matter

of convenience.

Show that ifnis a positive integer

Γ(n, x)=(n−1)!e−x

∑n−^1

k=0

xk

k!

.

From (18.166), on integrating by parts we find

Γ(n, x)=

∫∞

x

un−^1 e−udu=xn−^1 e−x+(n−1)

∫∞

x

un−^2 e−udu

=xn−^1 e−x+(n−1)Γ(n− 1 ,x),

which is valid for arbitraryn.Ifnis an integer, however, we obtain

Γ(n, x)=e−x[xn−^1 +(n−1)xn−^2 +(n−1)(n−2)xn−^3 +···+(n−1)!]

=(n−1)!e−x

∑n−^1

k=0

xk

k!

,