Suppose $f(x) = ax^{-q}$ and $g(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp[{\frac{-1}{2\sigma^2}(x)^2}]$.
I am attempting to find the convolution $(f\star g)(r) = \int\limits_{0}^{\infty}dx \ ax^{-q} \frac{1}{\sqrt{2\pi}\sigma} [\exp{\frac{-1}{2\sigma^2}(r-x)^2}]$.
Assuming $q>0$. I have been searching for a way to solve this analytically (even with mathematica), but I cannot seem to find a solution for $q>0$. Is there any way to go about writing down an analytical solution to such an integral? Thanks.
I suppose that there is a serious problem close to $x=0$ since
$$ \frac{a}{\sqrt{2\pi}\sigma}\,x^{-q}\, \exp\Big[-\frac{(r-x)^2}{2\sigma^2}\Big]\sim\frac{a e^{-\frac{r^2}{2 \sigma ^2}}}{\sqrt{2 \pi } \sigma }x^{-q}$$
The problem is totally different and does not make any trouble if $q \leq 0$.