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Math Help - Absolute convergence of the beta function

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    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?
    Last edited by ProofbyInduction; April 13th 2012 at 06:44 AM.
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