# Evaluating a double integral containing absolute value squared of sum of two functions.

I need to evaluate integrals of this form

$\int\limits_{\phi = 0}^{2\pi }\ \int\limits_{\theta = 0}^{\pi} \left | f_1(\theta, \phi)e^{ix} + f_2(\theta, \phi)e^{ix'} \right |^2\sin{\theta} \ d\theta d\phi$

Where,

$f_1$ and $f_2$ are harmonic functions of $\theta$ and $\phi$. Mostly, $\cos^2{\frac{\theta}{2}}$ or $\sin^2{\frac{\theta}{2}}$ and in general, $f_1 \neq f_2$. $x$ and $x'$ are some real constants.

Any help regarding how I can evaluate these integrals is highly appreciated.

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Hint: If $f_1$ and $f_2$ are real, then $$|f_1 e^{i x}+f_2 e^{i x'}|^2 = f_1^2+2f_1 f_2\cos(x-x') + f_2^2.$$

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exactly what I was looking for! Was able to solve the integrals with this. Thanks :) – user5198 May 28 '13 at 4:45
@user5198: Glad to help. I thought this might crack it for you. – user26872 May 28 '13 at 4:51