# Integral of $\cos(\theta_1-\theta_2)/\sqrt{1-\cos(\theta_1-\theta_2)}$

I am trying to see if the following integral converges, and if it has a simple solution

$$\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{\cos(\theta_1-\theta_2)}{\sqrt{1-\cos(\theta_1-\theta_2)}}d\theta_1d\theta_2$$.

So far, the integral is eluding me. At first I thought about writing it as a Fourier series, but this does not help me solve the integral.

Any help or guidance would be appreciated.

• The integral divereges. Mar 26 '21 at 14:24

Mathematica says the integral converges to $$(2\sqrt{2}\pi^2)i$$. So it probably involves a complex substitution. I'm working on it. I will let you know if I figure it out.