Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

If we letx=n+y,then

lnx=lnn+ln

(

1+

y

n

)

=lnn+

y

n

−

y^2

2 n^2

+

y^3

3 n^3

−···.

Substituting this result into (18.161), we obtain

n!=

∫∞

−n

exp

[

n

(

lnn+

y

n

−

y^2

2 n^2

+···

)

−n−y

]

dy.

Thus, whennis sufficiently large, we may approximaten!by

n!≈enlnn−n

∫∞

−∞

e−y

(^2) /(2n)
dy=enlnn−n

√

2 πn=

√

2 πn nne−n,

which is Stirling’s approximation (18.160).

18.12.2 The beta function

Thebeta functionis defined by

B(m, n)=

∫ 1

0

xm−^1 (1−x)n−^1 dx, (18.162)

which converges form>0,n>0, wheremandnare, in general, real numbers.

By lettingx=1−yin (18.162) it is easy to show thatB(m, n)=B(n, m). Other

useful representations of the beta function may be obtained by suitable changes

of variable. For example, puttingx=(1+y)−^1 in (18.162), we find that

B(m, n)=

∫∞

0

yn−^1 dy

(1 +y)m+n

. (18.163)

Alternatively, if we letx=sin^2 θin (18.162), we obtain immediately

B(m, n)=2

∫π/ 2

0

sin^2 m−^1 θcos^2 n−^1 θdθ. (18.164)

The beta function may also be written in terms of the gamma function as

B(m, n)=

Γ(m)Γ(n)

Γ(m+n)

. (18.165)

Prove the result (18.165).

Using (18.157), we have

Γ(n)Γ(m)=4

∫∞

0

x^2 n−^1 e−x

2

dx

∫∞

0

y^2 m−^1 e−y

2

dy

=4

∫∞

0

∫∞

0

x^2 n−^1 y^2 m−^1 e−(x

(^2) +y (^2) )
dx dy.