How can I solve this integral involving the error function?

[omyojj]

could anyone give me a hint to calculate this integral?

integral _{-inf, +inf} { exp(-x^2) / (x^2 + a^2) } _ dx

(I`m ignorant of tex)

the answer given from the mathematica is e^(a^2)/a * Pi * Erfc[a]

but there is no process of detailed calculation..

please give me a hand..

 

[HallsofIvy]

"Erfc" itself cannot be written in terms of "elementary functions"

 

[omyojj]

re

[tex]\int -\infty^\infty frac{e^{-x^2}{x^2+a^2}dt[\tex]

 

[omyojj]

sorry..Now I can type LaTex a little

I think that one of the possible ways to get the right answer is..

$$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{x^2+a^2} dx = 2 \int_{0}^{\infty} \frac{e^{-x^2}}{x^2+a^2} dx = 2e^{a^2} \int_{a}^{\infty} \frac{e^{-x^2}}{x\sqrt{x^2-a^2}}$$

by substituting x^2 by x^2+a^2. Perhaps we will need formulae
$$\begin{multline*}\frac{d}{dx}\mathrm{erf}(x) = e^{-x^2} \\ \frac{d}{dx}[-\frac{1}{a}\arctan(\frac{a}{\sqrt{x^2-a^2}})]=\frac{1}{x\sqrt{x^2-a^2}}\end{multline*}$$

But I cannot proceed further..