I Gaussian Quadrature on a Repeated Integral

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1. Nov 28, 2017

olukelliot

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

2. Nov 28, 2017

mathman

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.

3. Nov 28, 2017

eys_physics

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.