# Integrals with error function and exponentials

I'm trying to solve the integrals below:

$$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{x}{\sqrt{x^2+y^2}}\cdot \operatorname{erf}\left(m\cdot\sqrt{x^2+y^2}\right) \cdot \exp(-a\cdot (x-b)^2-a\cdot (y-c)^2)\mathrm dx\mathrm dy$$ where $a,b,c$ and $m$ are constants. Does anybody know how to solve the integral? Thanks!

• I edited your post to make it readable, please check if this fits your question. – Hirshy Aug 27 '15 at 8:32
• Yes, it is exactly my integral. Thanks! – snow Aug 27 '15 at 9:20
• with $b=c=0$ probably yes (0!) otherwise i would be really surprised – tired Aug 27 '15 at 11:18
• However, b and c are not 0. – snow Aug 27 '15 at 11:26
• First of all, you should differentiate with regard to the parameter m, to get rid of the $\dfrac1{\sqrt{x^2+y^2}}$ in the integrand. Secondly, the very presence of $\sqrt{x^2+y^2}$ in the exponent begs for a trigonometric or hyperbolic substitution. Unfortunately, the only functions which are defined in this way are the Bessel and Struve functions. – Lucian Aug 27 '15 at 14:08