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I am trying to make progress on the following integral

[itex] 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 [/itex]

where * denotes complex conjugate and the Fourier coefficients [itex]\alpha_n[/itex] 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 with out 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