Mathematical Methods for Physicists: A concise introduction - Site Map

THE GAMMA AND BETA FUNCTIONS
When ˆ n, a positive integer, repeated application of Eq. (2.40) and use of Eq.
(2.39) gives
…n ‡ 1† ˆn…n† ˆn…n 1†…n 1† ˆ...ˆ n…n 1†3 2 …1†
ˆ n…n 1†3 2 1 ˆ n!:
Thus the gamma function is a generalization of the factorial function. Eq. (2.40)
enables the values of the gamma function for any positive value of to be
calculated: thus
… 7 2 †ˆ…5 2 †…5 2 †ˆ…5 2 †…3 2 †…3 2 †ˆ…5 2 †…3 2 †…1 2 †…1 2 †:
p
Write u ˆ‡
x in Eq. (2.38) and we then obtain
so that
…† ˆ2
Z 1
0
0
u 21 e u2 du;
Z 1
… 1 2 †ˆ2 p
e u2 du ˆ :
The function …† has been tabulated for values of between 0 and 1.
When 0†: …2:41†
Substituting t ˆ v=…1 ‡ v†, this can be written in the alternative form
B…p; q† ˆ
Z 1
0
v p1 …1 ‡ v† pq dv:
…2:42†
By writing t 0 ˆ 1 t we deduce that B…p; q† ˆB…q; p†.
The beta function can be expressed in terms of gamma functions as follows:
B…p; q† ˆ…p†…q†
…p ‡ q† :
…2:43†
To prove this, write x ˆ at (a > 0) in the integral (2.38) de®ning …†, and it is
straightforward to show that
…†
a
ˆ
Z 1
0
95
e at t 1 dt
…2:44†