I am working on a challenging multiple integral problem and would appreciate any assistance. The integral is as follows:
$$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \ldots \int_{-\infty}^{+\infty} \log(\sqrt{x_1^2+y_1^2}) \cdot \log(\sqrt{x_2^2+y_2^2}) \cdot \ldots \cdot \log(\sqrt{x_m^2+y_m^2}) \cdot \delta\left(\sum_{i=1}^{N} (x_i^2+y_i^2)-1\right) \, dx_1 \, dy_1 \, dx_2 \, dy_2 \ldots dx_N \, dy_N$$
where $ m $ is less than $ N $, and $ N $ is a large natural number. The Dirac delta function ($ \delta $) imposes a constraint on the problem.
I have seen some of your insightful answers on related topics, @achille hui, and I was wondering if you could offer any guidance or suggestions on this problem. Your expertise would be greatly appreciated.
Thank you,