Gaussian Quadrature on a Repeated Integral

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Discussion Overview

The discussion revolves around the evaluation of a repeated integral using Gaussian Quadrature. Participants explore the feasibility of applying this numerical method to the integral and address potential ambiguities and singularities in the formulation.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses difficulty in evaluating a repeated integral and questions the applicability of Gaussian Quadrature for this purpose.
  • Another participant points out the ambiguity in the notation used, specifically the use of φ for both integration variables and the integrand, suggesting the need for distinct symbols.
  • A further contribution highlights the potential for the integral to be singular, depending on the unspecified values of ##r_1##, ##m##, and ##l##.
  • Another participant suggests splitting the repeated integral into at least two simpler integrals with different limits, which could then be numerically integrated using Gaussian Quadrature.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the method for evaluating the repeated integral, with multiple viewpoints and suggestions presented without resolution.

Contextual Notes

The discussion reveals limitations related to the ambiguity in variable notation and the unspecified parameters that may affect the behavior of the integral.

olukelliot
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upload_2017-11-28_18-6-45.png

Hi there,
I am having some difficulty evaluating a repeated integral, which is the first of two shown in the image.
I had hoped to be able to use Gaussian Quadrature to provide a numerical result, however am unsure on if this is possible for a repeated integral?

I have attempted to use Cauchy' formula on repeated integrals to obtain a single integral, which is shown on the bottom in the image. However I am once again unsure on performing this due to the presence of φ.

Any ideas on what I'm doing wrong/ missing?
Thanks
 

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Your first integral is too ambiguous. You use φ for both integrations as well as as a variable in the integrand. You need to use two different symbols for the differential variables, so there would be no ambiguity for the variable in the integrand.
 
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Additionally, it seems like the integral could be singular, depending on the values of ##r_1##, ##m## and ##l##. It is not clear from your thread what the values of these quantities are.
 
You have to split the repeated integral into min. two simple integrals (with different borders) and each of them could be numerical integrated by Gauss.
See:
COMPUTATION OF DEFINITE INTEGRAL OVER REPEATED INTEGRAL Katar´ina Tvrda´, Maria Minarova´
Tatra mountains matematical publications
 

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