Solving Difficult Integral: \int_{-\infty}^{+\infty}Exp(-x^2)*Erf(x^2 - a^2)dx

[Heimdall]

Hi,

I have an integral that I find quite difficult, I can't obtain anything from mathematica (but I'm far from being an expert).

Would some of you have a hint ? is it analytical ? It seems to be a "simple" function, from the physicist I am it should be integrable...

this integral is :

$$\int_{-\infty}^{+\infty}Exp(-x^2)*Erf(x^2 - a^2)dx$$Thanks !

 

[CompuChip]

Mathematica doesn't give me an analytical solution either.
But the integrand seems to be sharply peaked around x = 0, and we have the nice result
$$\frac{d}{dx} \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} e^{-x^2}$$
so maybe you can do something like steepest descent (expand the integrand around 0)?