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Double Integral: solution with hypergeometric function?

  1. Mar 5, 2017 #1
    1. The problem statement, all variables and given/known data
    Hello, I've recently encountered this double integral
    $$\int_0^1 dv \int_0^1 dw \frac{(vw)^n(1-v)^m}{(1-vw)^\alpha} $$

    with ## \Re(n),\Re(m) \geq 0## and ##\alpha = 1,2,3##.

    2. Relevant equations

    I use Table of Integrals, Series and Products by Gradshteyn & Ryzhik as a reference. There I found that
    $$\int_0^1 dv \frac{(v)^n(1-v)^m}{(1-vw)^\alpha} = \beta(n+1, m+1)
    _2F_1(\alpha ,n+1; n+m+2; w) \qquad\text{(3.197.3)} $$
    with the ##\beta##-function and ##_2F_1## the ordinary hypergeometric function. But this equation is only valid for ##w<1## (and ## \Re(n+1),\Re(m+1) > 0##, which is the case here).

    3. The attempt at a solution
    If I could use the formula above, I could easily integrate over ##v## and then over ##w##, but I suppose it would be wrong as ##w=1## at the upper integration limit. Can I somehow bypass this?
    Additionnally if we were to only consider ##w=1##, the integral would diverge at least in some cases (e.g. ##m=0##). Could my whole integral diverge because of this?
     
  2. jcsd
  3. Mar 6, 2017 #2

    DrDu

    User Avatar
    Science Advisor

    At w=1, the integral from Gradstein diverges logarithmically. But the integral over a logarithmic singularity is finite, so the integral over w should exist.
     
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