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Absolite convergence of the beta function

  1. Apr 13, 2012 #1
    Determine the values of the complex parameters [itex]p[/itex] and [itex]q[/itex] for which the beta function [tex]\int_0^1 t^{p-1} (1-t)^{q-1}dt[/tex] converges absolutely.

    The solution says:

    Split the integral into 2 parts: [tex]\displaystyle \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 [/tex]

    As [itex]t \to 0[/itex] [tex]|t^{p-1} (1-t)^{q-1}| \sim |t^{p-1}| \Rightarrow \text{Re}(p)>0[/tex] As [itex]t\to 1[/itex] [tex]|t^{p-1} (1-t)^{q-1}| \sim |(1-t)^{q-1}| \Rightarrow \text{Re}(q)>0[/tex]

    Why are these implications true?
    Last edited: Apr 13, 2012
  2. jcsd
  3. Apr 13, 2012 #2
    because otherwise a neighborhood of the endpoints is non-integrable?
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