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Error function integral

  1. Apr 19, 2008 #1
    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. jcsd
  3. Apr 19, 2008 #2


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    "Erfc" itself cannot be written in terms of "elementary functions"
  4. Apr 19, 2008 #3

    [tex]\int -\infty^\infty frac{e^{-x^2}{x^2+a^2}dt[\tex]
    Last edited: Apr 19, 2008
  5. Apr 19, 2008 #4
    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
    [tex]\begin{multline*}\frac{d}{dx}\mathrm{erf}(x) = e^{-x^2} \\

    But I cannot proceed further..
    Last edited: Apr 19, 2008
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