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Definite integral of exp and error function

  1. Jun 11, 2012 #1

    I've been trying to evaluate the following integral

    [tex] \int_{z}^{\infty}\exp\left(-y^{2}\right)\mathrm{erf}\left(b\left(y-c\right)\right)\,\mathrm{d}y [/tex]

    or equivalently

    [tex] \int_{z}^{\infty}\exp\left(-y^{2}\right)\mathrm{erfc}\left(b\left(y-c\right)\right)\,\mathrm{d}y [/tex]

    [tex]\mathrm{erf}\left(x\right)=\frac{2}{\sqrt{\pi}} \int_{0}^{x}\exp\left(-u^{2}\right)\,\mathrm{d}u, \quad\quad \mathrm{erfc}\left(x\right)=1-\mathrm{erf}\left(x\right)=\frac{2}{\sqrt{\pi}} \int_{x}^{+\infty}\exp\left(-u^{2}\right)\,\mathrm{d}u[/tex]

    I guess I tried to employ all techniques I'm familiar with but with no result.
    Can anyone help me with this one, please?
    Thank you!
    Last edited: Jun 11, 2012
  2. jcsd
  3. Jun 11, 2012 #2


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    To the best of my knowledge, it can't be done analytically. I suggest you start with the erf representation and see if the two exponentials might be combined into one, so that you might have an erf for the integral.
  4. Jun 15, 2012 #3
    Thanks mathman for your reply. I guess I'm not able to deal with this integral. I have a question though. I'm not a mathematician nor a math student so I was wondering if anyone could explain to me why the integral

    [tex]\int_{-\infty}^{\infty}\exp\left(-y^{2}\right) \mathrm{erf}\left(b\left(y-c\right)\right)\,\mathrm{d}y=-\sqrt{\pi}\,\mathrm{erf}\left(\frac{bc}{\sqrt{1+b^{2}}}\right)[/tex]

    can be evaluated quite easily (using differentiation under integral sign method) and the integral from my original post seems to have no analytical solution?

  5. Jun 15, 2012 #4


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    I haven't looked at it in detail, but it looks like the problem is analogous to integrating the Gaussian. When you integrate over the entire real line you get a neat analytic solution, but integrating over part of the line ends up with erf.
  6. Jun 15, 2012 #5
    Ok, I guess I know what you mean. Thanks again!
  7. Jun 16, 2012 #6
    Hi !

    in attachment, a method for solving the definite integral.

    Attached Files:

  8. Jun 17, 2012 #7
    Hi JJacquelin! Your post helped me with showing that the constant of integration [tex]C=0[/tex] in a more general formula:

    \int_{-\infty}^{\infty}\exp\left(-b^{2}(x-c)^{2}\right)\mathrm{erf}\left(a\left(x-d\right)\right)\,\mathrm{d}x=\frac{\sqrt{\pi}}{b} \mathrm{erf}\left(\frac{ab\left(c-d\right)}{\sqrt{a{}^{2}+b{}^{2}}}\right),\quad b>0

    Thank you!
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