Integral: square root of sum of trig polynomials

In summary, the conversation discusses a difficult integral involving complex coefficients and a square root. The person is struggling to find an analytical approach and may resort to numerical integration. They also mention using the expression in a Lagrangian.
  • #1
nickthequick
53
0
Hi,

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 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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  • #2
Definitely numerically. You try it analytically, you'll wind up in an asylum somewhere.
 
  • #3
Hm. I suspected as much. I'm going to end up differentiating this term w.r.t [itex]\alpha_n[/itex] 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
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.
 

What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is the reverse operation of differentiation, which is finding the slope of a curve at a given point.

What is the square root of a sum of trigonometric polynomials?

The square root of a sum of trigonometric polynomials is a mathematical expression that involves taking the square root of a sum of multiple trigonometric functions, such as sine, cosine, and tangent. It is often used in solving equations and finding areas under curves.

How is the integral of a square root of a sum of trigonometric polynomials calculated?

The integral of a square root of a sum of trigonometric polynomials is calculated using integration techniques, such as substitution, integration by parts, or trigonometric identities. It is important to consider the domain of the function and use the appropriate integration method.

What are the applications of the integral of a square root of a sum of trigonometric polynomials?

The integral of a square root of a sum of trigonometric polynomials has various applications in fields such as physics, engineering, and economics. It is used to calculate areas, volumes, and moments of inertia, among others.

What are some common mistakes when solving integrals of square roots of sum of trigonometric polynomials?

Some common mistakes when solving integrals of square roots of sum of trigonometric polynomials include forgetting to consider the domain of the function, using incorrect integration techniques, and making errors in simplifying trigonometric identities. It is important to double-check the solution and be familiar with integration rules and trigonometric properties.

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