Error function integral

  • Thread starter omyojj
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Main Question or Discussion Point

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..
 

Answers and Replies

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

PHP:
[tex]\int -\infty^\infty frac{e^{-x^2}{x^2+a^2}dt[\tex]
 
Last edited:
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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}}
[/tex]

by substituting x^2 by x^2+a^2. Perhaps we will need formulae
[tex]\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*}[/tex]

But I cannot proceed further..
 
Last edited:

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