- #1

omyojj

- 37

- 0

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..

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- Thread starter omyojj
- Start date

- #1

omyojj

- 37

- 0

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..

- #2

HallsofIvy

Science Advisor

Homework Helper

- 43,021

- 970

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

- #3

omyojj

- 37

- 0

PHP:

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

Last edited:

- #4

omyojj

- 37

- 0

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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