Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to solve an integral with the Inverse error function

  1. May 16, 2014 #1
    this is not a homework and my problem is much bigger for me to give full details here. I came across this integral

    [itex]\mathcal{I}(\xi)=\int^{\xi_c}_{\xi}{\rm d}\xi^\prime\exp\left[\sqrt{2}\sigma\,{\rm Erf}^{-1}\left(1-\frac{8\pi}{3}{\xi^\prime}^3\right)\right][/itex]

    where Erf[itex]^{-1}[/itex] is the inverse error function and

    [itex]\xi_c=\left[\frac{3}{8\pi}\left(1-{\rm Erf}\left(\frac{\sigma^2-\sqrt{2}\sigma\,{\rm Erf^{-1}(2\beta-1)}}{\sqrt{2}\sigma}\right)\right)\right]^{1/3}[/itex]

    with [itex]0\le\beta\le1[/itex].

    I would like get an analytical approximation but I can't figure out a way to do that, even with software like Mathematica. I tried solving the integral numerically and I find a reliable solution, however, I'm mostly interested in points where [itex]\xi\to\xi_c^-[/itex], and here the inverse error function diverges.

    Do you have any ideas on how to approximate this integral?
  2. jcsd
  3. May 17, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    Use a taylor series?
  4. May 17, 2014 #3
    Thank you for your answer. But I don't understand how I can Taylor expand where the integrand diverges.
    Also, do you see a way to normalize the integral and/or make it more simple to solve it numerically.
    What I find challenging is that for every value of beta I have to numerically solve the integral. It would be better to normalize somehow the result with beta and then do the integral only once.
  5. May 17, 2014 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    The inverse errorfunction can be defined as a Maclaurin Series.
    That seems like a way to get rid of some awkwardness there, but I don't know for sure that it is the best way to go.

    But I don't immediately see how to make it much simpler besides that.
  6. May 17, 2014 #5
    Thanks, the problem seems very complicated and I think I have to resort to a numerical integration.
    However, that would be easier if one could write


    so this way I would have to integrate only once. Do you see a way to possibly achieve that?
  7. Feb 2, 2017 #6
    I understand this is quite an old post but: any luck with computing this integral? I have come across something similar recently.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted