The improper integral is $$ \int_{-\infty}^{\infty}\frac{1}{\sqrt{y^2-t^2}}\cdot\exp{\left[ -\frac{(x-t)^2}{A} \right]} ~\mathrm{d}t = \int_{-\infty}^{\infty}\left( y^2-t^2 \right)^{(-1/2)}\cdot\exp{\left( \frac{-x^2-t^2}{A} \right)} \cdot \exp(2xt/A) ~\mathrm{d}t $$ and I want to simplify it.
I notice it's similar to the gamma function $\int_{0}^{\infty}\mathrm{e}^{-t}t^{z-1} ~\mathrm{d}t$, but I can't handle the term $\exp(2xt/A)$.