# Error function integral

1. Apr 19, 2008

### 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..

plz give me a hand..

2. Apr 19, 2008

### HallsofIvy

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

3. Apr 19, 2008

re

PHP:
$$\int -\infty^\infty frac{e^{-x^2}{x^2+a^2}dt[\tex] Last edited: Apr 19, 2008 4. Apr 19, 2008 ### omyojj sorry..Now I can type LaTex a little I think that one of the possible ways to get the right answer is.. [tex] \int_{-\infty}^{\infty} \frac{e^{-x^2}}{x^2+a^2} dx = 2 \int_{0}^{\infty} \frac{e^{-x^2}}{x^2+a^2} dx = 2e^{a^2} \int_{a}^{\infty} \frac{e^{-x^2}}{x\sqrt{x^2-a^2}}$$

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..

Last edited: Apr 19, 2008