Problems on beta & gamma functions ? Please help & explain?

Question

Q1 Prove the following
(i) 0 to infinity , Integration of [e ^ -(x^.5)] / x ^ (7/4) dx = 8/3 (pie ^ .5)

(ii) 0 to 1 , Integration of (y ^ q-1) [(log x ^ -1) ^ p-1] dy = [Gamma (p) ] / q ^ p , where p>0,q>0

(iii) 0 to 1 , Integration of (x ^ m) [(log x)] ^ n dx =
[(-1) ^ n !] / (m+1) ^ n+1 , where n is a postive integer and m >-1

(iv) Given 0 to infinity , Integration of (x ^ n-1) / (1+x) dx =
pie / sin( n pie) , prove that [gamma (n)] [gamma (1-n)] =
pie/ sin (n pie)

Q2 Express 0 to 1 , Integration of (x ^ m) [(1-x ^ n ) ^ p] dx in terms of gamma functions .

note :- ! is the factorial sign .

Answer 1

HOMEWORK!

Hints:
Q1 i) Let x=u^2, so dx=2u du to give
int_0^infty e^(-u) u^(7/2) 2u du=2Gamma(7/2)

ii) I think you want log y, not log x. Let u=-log y, so y=e^-u.

iii) Let x=e^u

iv) You need restrictions on n for this to be true, at least if you use the integral form for the Gamma function. By analyticity, it is always true when both sides are defined.

Q2. Look at Bessell functions and use u=x^n.

Answer 2

How to write all the notations here ??????????