Page 400 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 15] GAMMA AND BETA FUNCTIONS 391

15.72. Obtain (12)on Page 378 from the result (4)of Problem 15.20.
3 p ffiffi
[Hint: Expand e v =ð3 nÞ þ in a power series and replace the lower limit of the integral by 1.]
15.73. Obtain the result (15)on Page 378.
1
ðx þ !Þ,thus ln ðxÞ¼ ln ðx þ 1Þ ln x, and
x
[Hint: Observe that ðxÞ¼
1
0 0
ðxÞ ðx þ 1Þ
¼
x
ðxÞ ðx þ 1Þ
Furthermore, according to (6) page 377.
k! k x
ðx þ !Þ¼ lim
k!1 ðx þ 1Þ ðx þ kÞ
Now take the logarithm of this expression and then differentiate. Also recall the definition of the Euler
constant,
.
15.74. The duplication formula (13a) Page 378 is proved in Problem 15.24. For further insight, develop it for
positive integers, i.e., show that
1
ffiffiffi
2 2n 1 ðn þ Þ ðnÞ¼ ð2nÞ
p
2
1
Hint: Recall that ð Þ¼ ,thenshow that
2

1
:
ðn þ Þ¼ 2n þ 1Þ ð2n 1Þ 5 3 1 p ffiffiffi
2 2 ¼ 2 n
Observe that
ð2nÞ!
ð2n þ 1Þ ¼ð2n 1Þ 5 3 1
n
n ¼ 2 n!
2 ðn þ 1Þ
Now substitute and refine.