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Showing Complete Elliptic Integral of First Kind Maps to Rectangle

  1. Dec 10, 2011 #1
    1. The problem statement, all variables and given/known data

    Effectively, I'm trying to show the following two integrals are equivalent:
    [tex]\int_1^{1/k}[(x^2-1)(1-k^2x^2)]^{-1/2} dx = \int_0^1[(1-x^2)(1-(k')^2x^2)]^{-1/2}dx[/tex]
    where [itex]k'^2 = 1-k^2[/itex] and [itex]0 < k,k' < 1[/itex].

    2. Relevant equations

    One aspect of the problem I showed the following:
    [tex]\int_{1/k}^\infty [(x^2-1)(k^2x^2-1)]^{-1/2}dx = \int_0^1 [(1-x^2)(1-k^2x^2)]^{-1/2}dx[/tex]
    However, I'm not entirely sure that this is needed.

    3. The attempt at a solution
    I've been told that I should use the substitution [itex]u = (1-k'^2 x^2)^{-1/2}[/itex] in order to prove this relation, but while I do in fact get the correct denominator in my integral, there is a constant out front in terms of the ks and the limits of integration are not of the form I want, and I'm having difficulties determining where to go from there without altering the denominator's form. When I try to use different but similar substitutions, while I can get proper limits of integration I lose out on the term in the integrand.

    Any suggestions on how to proceed from there would be helpful.
    Last edited: Dec 11, 2011
  2. jcsd
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