I am trying to find the integral of
\begin{align} f(y) &= \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} e^{- a \left|| x - y \right||^2} dx \, dy \\ &= \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{y} e^{a \left( y - x \right)^2} dx \, dy + \int\limits_{-\infty}^{\infty} \int\limits_{y}^{\infty} e^{a \left( x - y \right)^2} dx \, dy \end{align}
but I'm having trouble finding the solution to each part. I'd really appreciate help if anyone knows how to do it? I'm guessing there's an $\operatorname{erf}()$ function in there somewhere.
Thanks in advance for your help.
Cheers,
Katie
