Page 387 - Schaum's Outline of Theory and Problems of Advanced Calculus

378 GAMMA AND BETA FUNCTIONS [CHAP. 15

Now use (8)to produce
ðxÞ ð1 xÞ¼
( ) ! !
1
1
1 Y 1 Y
1 x=k
Y
2
e
e
x 1
x lim ð1 þ x=kÞ e e
x lim ð1 x=kÞ 1 x=k ¼ lim ð1 ðx=kÞ Þ
¼1 ¼1 ¼1
k!1 k!1 x k!1
Thus

; 0 < x < 1
sin x
ðxÞ ð1 xÞ¼ ð11aÞ
Observe that (11) yields the result
1 p ffiffiffi
2
ð Þ¼ ð11bÞ
Another exact representation of ðx þ 1Þ is

ffiffiffiffiffiffi 1 1 139
p xþ1 x
2 x e
ðx þ 1Þ¼ 1 þ þ 2 þ 3 þ ð12Þ
12x 288x 51840x
The method of obtaining this result is closely related to Sterling’s asymptotic series for the gamma
function. (See Problem 15.20 and Problem 15.74.)
The duplication formula
2 2x 1 1 p ffiffiffi
2 ¼ ð2xÞ ð13aÞ
ðxÞ x þ
also is part of the literature. Its proof is given in Problem 15.24.
The duplication formula is a special case ðm ¼ 2Þ of the following product formula:

1 2 m 1 1 mx m 1
¼ m 2
m m m
ðxÞ x þ x þ X þ ð2 Þ 2 ðmxÞ ð13bÞ
It can be shown that the gamma function has continuous derivatives of all orders. They are
obtained by differentiating (with respect to the parameter) under the integral sign.
ð
1 x 1 yt x 1 x 1
t e dt and that if y ¼ t , then ln y ¼ ln t ¼ðx 1Þ ln t.
It helps to recall that ðxÞ¼
1 0
Therefore, y ¼ ln t.
0
y
It follows that
ð
1
x 1 t
t e ln tdt:
0
ðxÞ¼ ð14aÞ
0
This result can be obtained (after making assumptions about the interchange of differentiation with
limits) by taking the logarithm of both sides of (9) and then differentiating.
In particular,
ð1Þ¼
ð
is Euler’s constant.) ð14bÞ
0
It also may be shown that

1 1 1 1 1 1
0
ðxÞ
1 x 2 x þ 1 n x þ n 1
¼
þ þ þ ð15Þ
ðxÞ
(See Problem 15.73 for further information.)
THE BETA FUNCTION
The beta function is a two-parameter composition of gamma functions that has been useful enough in
application to gain its own name. Its definition is