# Showing Complete Elliptic Integral of First Kind Maps to Rectangle

1. Dec 10, 2011

### Cider

1. The problem statement, all variables and given/known data

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

2. Relevant equations

One aspect of the problem I showed the following:
$$\int_{1/k}^\infty [(x^2-1)(k^2x^2-1)]^{-1/2}dx = \int_0^1 [(1-x^2)(1-k^2x^2)]^{-1/2}dx$$
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 $u = (1-k'^2 x^2)^{-1/2}$ 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