# How to prove these integral identities involving erf

Can anybody tell me how to prove the following integral identities (1) and (2) involving error function erf?

(1) $$\int_{0}^{t}\int_{0}^{\infty}\frac{\exp\left[-\frac{\left(x-\xi\right)^{2}}{4D\left(t-\tau\right)}\right]-\exp\left[-\frac{\left(x+\xi\right)^{2}}{4D\left(t-\tau\right)}\right]}{\sqrt{4\pi D\left(t-\tau\right)}}erf\left(\frac{\xi}{\sqrt{4D\tau}}\right)\, d\xi\, d\tau=t\, erf\left(\frac{x}{\sqrt{4Dt}}\right)$$.

It is assumed that $$D>0$$ and $$x\geq0$$, and the internal integration is along $$\xi$$. MATHEMATICA function Integrate[] fails to calculate the above integral symbolically (there is a question why?), but numerical evaluation confirms the identity.

I would also be happy to be able to calculate analytically (or prove that this is impossible) the integral $$\int_{0}^{t}\int_{0}^{\infty}\frac{\exp\left[-\frac{\left(x-\xi\right)^{2}}{4D\left(t-\tau\right)}\right]-\exp\left[-\frac{\left(x+\xi\right)^{2}}{4D\left(t-\tau\right)}\right]}{\sqrt{4\pi D\left(t-\tau\right)}}\left[erf\left(\frac{\xi}{\sqrt{4D\tau}}\right)\right]^{2}\, d\xi\, d\tau$$.

(2) $$\int_{0}^{\infty}\exp\left(-\xi^{2}a^{2}\right)erf\left(\xi b\right)\, d\xi=\frac{\arctan\left(b/a\right)}{a\sqrt{\pi}}$$,

where $$a>0$$ and $$b>0$$. This result is returned by MATHEMATICA, but I would be glad to know how it is obtained.

Leslaw

For (2), you can use the Maclaurin series for $$\text{erf}(\xi b)$$ and the fact that $$\int_0^\infty \exp(-\xi^2 a^2) \xi^{2k+1}\; d\xi = \frac{k!}{2 a^{2k+2}}$$