Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS

Γ( )

1 2

2

3 4

4

− (^2) − 1

− 2

− 4 − 3

− 4

− 6

6

n

n

Figure 18.9 The gamma function Γ(n).

Moreover, it may be shown for non-integralnthat the gamma function satisfies

the important identity

Γ(n)Γ(1−n)=

π

sinnπ

. (18.158)

This is proved for a restricted range ofnin the next section, once the beta

function has been introduced.

It can also be shown that the gamma function is given by

Γ(n+1)=

√

2 πn nne−n

(

1+

1

12 n

+

1

288 n^2

−

139

51 840n^3

+...

)

=n!,

(18.159)

which is known asStirling’s asymptotic series. For largenthe first term dominates,

and so

n!≈

√

2 πn nne−n; (18.160)

this is known asStirling’s approximation. This approximation is particularly useful

in statistical thermodynamics, when arrangements of a large number of particles

aretobeconsidered.

Prove Stirling’s approximationn!≈

√

2 πn nne−nfor largen.

From (18.153), the extended definition of the factorial function (which is valid forn>−1)
is given by

n!=

∫∞

0

xne−xdx=

∫∞

0

enlnx−xdx. (18.161)