# How to solve an integral with a Gaussian Mixture denominator?

I am trying to solve this integral:

$$t(v)\equiv\sum_{k=1}^{n}\sum_{j=1}^{n}\int_{-\infty}^{\infty}\frac{w_{k}N(x-x_{k},B_{k})N(x-x_{j},v)}{\sum_{m=1}^{n}w_{m}N(x-x_{m},B_{m}+v)}dx$$

where $\sum_{k=1}^{n}w_{k}=1$, $N(x,q)\equiv\frac{1}{\sqrt{2\pi q}}e^{-\frac{x^{2}}{2q}}$, {$v,B_1,...B_n,w_1,...w_n,$} are all real and positive, and {$x_1,...x_n$} are all real.

This integral can also be written as

$$t(v)\equiv\sum_{j=1}^{n}\int_{-\infty}^{\infty}\frac{f(x)N(x-x_{j},v)}{p(x,v)}dx$$

where $f(x)\equiv\sum_{k=1}^{n}w_{k}N(x-x_{k},B_{k})$ and $p(x,v)\equiv f(x)*N(x,v)$

Discussion:

1) The only way I could think of to solve or partially solve this is with Residue methods, but I welcome any method that works.

2) I'm not certain if the roots of a Gaussian Mixture of $n>2$ terms can be expressed analytically.

3) Given a root of $p[x,v]$, I'm not certain how to calculate the residue of the corresponding pole.

4) Given the infinite roots of $p[x,v]$ and the corresponding residues of the integrand, I'm not certain how to select a path of integration that will solve this problem.

5) Somewhere, at some point in my life, I have seen some problems of the form $\int_{-\infty}^{\infty}\frac{g(x)}{h(x)}dx$ where if certain relationships exist between $h(x)$ and $g(x)$, it was possible to solve the integral via residue-related methods without having to explicitly find the roots of $h(x)$. Can those methods be applied here? Can anyone point me to a reference to such methods if they recall them?

I suspect the result might come out involving $\sum_{j,k}\frac{N(x_{j}-x_{k},v)}{p(x_{k},v)}$ but I am not certain.

Thanks in advance for any help or insights.

• The first thought that came to mind when I saw this problem was to use a Pade approximation. Here's a link to some work I think might help. math.vanderbilt.edu/~esaff/texts/144.pdf Cheers!
– user277279
Oct 6, 2015 at 19:12