Edit: Oh, that's even better:
But the integrand is real-valued when . That doesn't make any sense, does it?
Oh, I thought a complex number was being squared, not necessarily a purely imaginary number. My bad.
Problem : For , compute
The answer is But the way I evaluate is to use the infinite product of Gamma function which makes me confused .
For this reason , i attempt to solve it without using the infinite product :
First , I need to obtain the expression of Digamma function :
Since substituting it we obtain :
Sub. we have
Let's get back to your problem :
Integrating it to obtain the result :
Using the similar method , Prove this formula ( Gauss's Multiplication Formula ) :
ps I don't know how many stars it should be given ...