# Is the integration of a gaussian function divided by polynomial possible?

I am trying to evaluate the following integral: $$$$I=\int_{-\infty}^{\infty}\exp\left \{-\frac{(u-1)^2}{2\sigma^2}\right\}\frac{1}{(u-x)^2+y^2}\mathrm{d}u$$$$ where $$x,y\in\mathbb{R}$$.

Trying on mathematica tells me that the integral does not converge. Moreover I cannot use the residue theorem due to the $$\exp\{-u^2\}$$ term. However when plotting the function for certain $$(x,y)$$ values the function seems to have a closed area so I would expect the integral to converge given some restraints on $$x$$ and $$y$$. How does one tackle such integrals?

In a broader context, under what conditions does the following integral converge? $$$$I_\text{gen}=\int_{-\infty}^{\infty}\exp\left \{-\frac{(x-\mu)^2}{2\sigma^2}\right\}\frac{1}{ax^2+bx+c}\mathrm{d}x$$$$

$$\DeclareMathOperator{\erfc}{erfc}$$ The integral \begin{align}\int_{-\infty}^{\infty}\frac{\mathrm{e}^{-t^2}\mathrm{d}t}{\left\lvert t- z\right\rvert^2}&=\frac{\pi}{\Im z}\Re\left(\mathrm{e}^{-z^2}\erfc(-\mathrm{i}z)\right) &&(\Im z \neq 0)\end{align} is standard (DLMF 7.19.4).
• Thanks, I did not know this identity! By rescaling, is the change of variables $u=\sqrt{2}\sigma t +1$ enough? It does take care of the exponential but then I get $|t\sqrt{2}\sigma+1-z|$ in the denominator. Sorry for asking something that is probably very simple, I just don't see it!
• pull out the factor of $\sqrt{2}\sigma$, then shift and rescale $z$ Commented Aug 24, 2019 at 16:52