Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

18.12.1 The gamma function

Thegamma functionΓ(n) is defined by

Γ(n)=

∫∞

0

xn−^1 e−xdx, (18.153)

which converges forn>0, where in generalnis a real number. Replacingnby

n+ 1 in (18.153) and integrating the RHS by parts, we find

Γ(n+1)=

∫∞

0

xne−xdx

=

[

−xne−x

]∞

0 +

∫∞

0

nxn−^1 e−xdx

=n

∫∞

0

xn−^1 e−xdx,

from which we obtain the important result

Γ(n+1)=nΓ(n). (18.154)

From (18.153), we see that Γ(1) = 1, and so, ifnis a positive integer,

Γ(n+1)=n!. (18.155)

In fact, equation (18.155) serves as a definition of the factorial function even for

non-integern. For negativenthe factorial function is defined by

n!=

(n+m)!

(n+m)(n+m−1)···(n+1)

, (18.156)

wheremis any positive integer that makesn+m>0. Different choices ofm

(>−n) do not lead to different values forn!. A plot of the gamma function is

given in figure 18.9, where it can be seen that the function is infinite for negative

integer values ofn, in accordance with (18.156). For an extension of the factorial

function to complex arguments, see exercise 18.15.

By lettingx=y^2 in (18.153), we immediately obtain another useful represen-

tation of the gamma function given by

Γ(n)=2

∫∞

0

y^2 n−^1 e−y

2

dy. (18.157)

Settingn=^12 we find the result

Γ

( 1

2

)

=2

∫∞

0

e−y

2

dy=

∫∞

−∞

e−y

2

dy=

√

π,

where have used the standard integral discussed in section 6.4.2. From this result,

Γ(n) for half-integralncan be found using (18.154). Some immediately derivable

factorial values of half integers are
(
−^32

)

!=− 2

√

π,

(

−^12

)

!=

√

π,

( 1

2

)

!=^12

√

π,

( 3

2

)

!=^34

√

π.