Integral: square root of sum of trig polynomials

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SUMMARY

The integral I = ∫₀²π √{(1 + ∑ₙ₌₁ᴺ αₙ e^{-inx})(1 + ∑ₙ₌₁ᴺ αₙ* e^{inx})} dx presents significant challenges due to the square root complicating the orthogonality of trigonometric polynomials. The discussion emphasizes that numerical methods are the most viable approach for evaluating this integral, as analytical methods may lead to complex difficulties. The user Nick is considering differentiating with respect to the coefficients αₙ, although this may not directly aid in solving the integral. Expanding the expression under the radical for potential cancellations is also suggested as a possible strategy.

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nickthequick
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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
 
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Definitely numerically. You try it analytically, you'll wind up in an asylum somewhere.
 
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.
 
If the an are unspecified, then numerical integration would also appear to be out the window as well.
The only other course is to expand the expression under the radical and hope for spit-load of cancellations.
 

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