Integral with some symmetries

In summary, the problem involves integrating a function with a nice structure, but a bound is needed if integration is not possible. The problem can be viewed as a minimization problem with n parameters. The provided link shows the necessary equations, including the use of distances and angles to calculate the function. The attempt at a solution involved trying to find a bound by using the modulus and the triangle inequality on the denominator, but this did not lead to a solution. Further insights or suggestions are requested to solve the integral or find a bound for its value.
  • #1

Homework Statement

Hi, i need to integrate a function which has a very nice structure. however, in case i cannot do the same, i would need a bound on the value of the integral. The problem is basically a minmization problem with n parameters. Please follow the link to acquaint your self with the problem. [Broken]

Homework Equations [Broken]
here, d_i and theta_i are the distances and angles of point (x,y) from (xi,yi)

The Attempt at a Solution

My attempt is as follows:
First of al, i could not directly integrate the function, so i tried to find a bound on it. I used the mod of the function (extension of triangle inequality) on the denominator to eliminate the sine squared in the denominator.
Upon doing that i tried to split the denominator through chebyshev sum inequality but didnt get anywhere.

I hope you can give me some insights into either solving the integral or getting a bound on its value.

thanks and regards
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  • #2
Would you elaborate on the geometry of the sum, specifically, where does the [tex]d_{i}^{-2}d_{j}^{-2}\sin^2 (\theta_{i}-\theta_{j})[/tex] term come from?

1. What is an integral with symmetries?

An integral with symmetries refers to an integral that has a certain level of symmetry in its integrand or integration limits. This means that the function being integrated or the boundaries of the integration exhibit some type of symmetry, making the integral easier to solve or evaluate.

2. How does symmetry affect integrals?

Symmetry can greatly simplify integrals by reducing the amount of computation required. If the integrand or integration limits are symmetric, the integral can often be rewritten in a simpler form or solved using symmetry properties such as even or odd functions. This can save time and effort in solving the integral.

3. What are some common examples of integrals with symmetries?

Some common examples of integrals with symmetries include integrals involving even or odd functions, integrals with symmetric integration limits (such as from -a to a), and integrals with circular or spherical symmetry. These types of integrals often have well-known solutions or can be easily evaluated using symmetry properties.

4. How can I identify symmetries in an integral?

To identify symmetries in an integral, look for patterns or properties in the integrand or integration limits. For example, if the integrand is an even function (f(-x) = f(x)), it will have reflective symmetry about the y-axis. If the integration limits are from -a to a, the integral will have rotational or point symmetry around the origin.

5. Are there any drawbacks to using symmetry in integrals?

While using symmetry can greatly simplify integrals, it is important to note that not all integrals have symmetries and relying solely on symmetry can limit the types of integrals that can be solved. Additionally, it is important to be aware of any restrictions or limitations that may arise when using symmetry, such as certain integration limits or functions that may not exhibit symmetries.

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