# Integral: square root of sum of trig polynomials

1. Oct 7, 2013

### 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 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

2. Oct 8, 2013

### SteamKing

Staff Emeritus
Definitely numerically. You try it analytically, you'll wind up in an asylum somewhere.

3. Oct 8, 2013

### nickthequick

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.

4. Oct 8, 2013

### SteamKing

Staff Emeritus
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.