# Integral: square root of sum of trig polynomials

nickthequick
Hi,

I am trying to make progress on the following integral

$I = \int_0^{2\pi} \sqrt{(1+\sum_{n=1}^N \alpha_n e^{-inx})(1+\sum_{n=1}^N \alpha_n^* e^{inx})} \ dx$

where * denotes complex conjugate and the Fourier coefficients $\alpha_n$ are constant complex coefficients, and unspecified. The square root throws off the ability to manipulate the orthogonality of the trig polynomials and I have been struggling to find a way to approach this problem without putting asymptotic bounds on the coefficients, which I do not want to do at this point.

Does anyone have any suggestions for how to attack this?

Thanks,

Nick

Staff Emeritus
Hm. I suspected as much. I'm going to end up differentiating this term w.r.t $\alpha_n$ as it is part of the potential term in a Lagrangian. I don't see how that would help solve this, but I note it for completeness.