# Absolute convergence of the beta function

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• Apr 13th 2012, 06:41 AM
ProofbyInduction
Absolute convergence of the beta function
Determine the values of the complex parameters $p$ and $q$ for which the beta function $\int_0^1 t^{p-1} (1-t)^{q-1}dt$ converges absolutely.

I can prove normal convergence when p and q are real:

Split the integral into 2 parts $\int_0^1 t^{p-1} (1-t)^{q-1}dt = \int_0^{1/2} t^{p-1} (1-t)^{q-1}dt + \int_{1/2}^1 t^{p-1} (1-t)^{q-1}dt$

Then for $t\in (0,1/2]$

$f(t) = t^{p-1} (1-t)^{q-1} \leqslant t^{p-1}$ and the integral of $t^{p-1}$ 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 $u=1-t$ 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:

$|f(t)| = |t^{p-1} (1-t)^{q-1}| \leqslant t^{p-1}$ and the integral of $t^{p-1}$ 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?