Determine the values of the complex parameters and for which the beta function converges absolutely.
I can prove normal convergence when p and q are real:
Split the integral into 2 parts
Then for
and the integral of converges if and only if p-1<1 (i.e. p>0). So the 1st integral converges if and only if p>0.
The same argument, letting gives that the 2nd integral converges if and only if q>0.
Now, does this still hold to prove absolute convergence and for complex p and q? So is it true that:
and the integral of converges if and only if Re(p-1)<1 (i.e. Re(p)>0). So the 1st integral converges absolutely if and only if Re(p)>0. Then the 2nd integral converges absolutely if and only if Re(q)>0.
If not how do I do it?