Integral: square root of sum of trig polynomials

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Discussion Overview

The discussion centers around the integral of the square root of a sum of trigonometric polynomials, specifically in the form of an integral involving complex coefficients. Participants explore methods to approach this integral without imposing asymptotic bounds on the coefficients.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Nick presents the integral I and expresses difficulty in manipulating it due to the square root, seeking suggestions for an approach.
  • One participant humorously suggests that numerical methods are the only viable option, implying that analytical attempts may lead to frustration.
  • Nick mentions a potential differentiation with respect to the coefficients as part of a Lagrangian, although he is uncertain about its usefulness in solving the integral.
  • Another participant notes that if the coefficients are unspecified, numerical integration may not be feasible and suggests expanding the expression under the square root to look for cancellations.

Areas of Agreement / Disagreement

Participants generally agree that the complexity of the integral makes analytical solutions challenging, with some advocating for numerical methods while others suggest alternative approaches like expansion. However, no consensus on a specific method has been reached.

Contextual Notes

Limitations include the unspecified nature of the coefficients, which complicates both numerical and analytical approaches. The discussion reflects uncertainty regarding the effectiveness of proposed methods.

Who May Find This Useful

Researchers or students interested in advanced integral calculus, particularly those dealing with complex coefficients and trigonometric polynomials.

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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