# Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe there is some way to simplify form or switch to spherical coordinates...

And here are steps which I performed by myself:

By integrating over φ, this can be reduced to one variable integral: Now I switch to complex plane: Now poles (there are 4, all lying on real axis, 2 of them are inside contour |z|=1: And suddenly, here comes sensation that there is root of polynomial, so there is branching at this points.

I've put this equation into Wolfram, but it returns something alienish: This integral on integrals.wolfram.com. Maybe it is possible to calculate all that limits at points θ=0 and θ=π. But there will be no way to prove that Mathematica solution, which is necessary for me :(

Thank you for your attention.

• By the time, I did nothing by expanding that Mathematica formula containing ellyptic function F(i*inf|m) in taylor series around m=1, resulting in (pi/sqrt(1-(a+b)^2))*(1+ab/4+...)... Maybe it's still unsolvalble in elementary functions. – Emil Dec 22 '13 at 22:59