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Error function and integrals

  1. Oct 31, 2011 #1
  2. jcsd
  3. Oct 31, 2011 #2

    Stephen Tashi

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    What constitutes an explanation will depend on your background. Do you know about integration techniques such a changing variables? And do you know that the definition of erfc(x) itself involves an integral?
  4. Oct 31, 2011 #3

    Char. Limit

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    Well, the definition of erfc(z) is actually:

    [tex]erfc(z) \equiv 1 - erf(z)[/tex]

    However, this can be pretty easily changed to the integral definition if we remember that

    [tex]erf(z) \equiv \frac{2}{\sqrt{\pi}} \int_0^z exp\left(-t^2\right) dt[/tex]

    and furthermore that

    [tex]\frac{2}{\sqrt{\pi}} \int_0^\infty exp\left(-t^2\right) dt = 1[/tex]

    By substituting these definitions in for 1 and erf(z), we get this expression:

    [tex]erfc(z) = \frac{2}{\sqrt{\pi}} \int_0^\infty exp\left(-t^2\right) dt - \frac{2}{\sqrt{\pi}} \int_0^z exp\left(-t^2\right) dt[/tex]

    Which is easily proven using basic integral properties to be equal to the expression in the OP.
  5. Nov 1, 2011 #4
  6. Nov 1, 2011 #5
    Thanks for the derivation. My final question is how one would analyze this integral (from 0 to z as you have it). Is it along the same lines we go about to derive the value of the error function from -inf to +inf?
  7. Nov 1, 2011 #6

    Char. Limit

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    We usually just find values numerically. The only value that I know of where erf(x) takes a simple, closed-form value is x=0. Other than that, it's all approximation.
  8. Nov 1, 2011 #7
    Thanks for your timely response. Although knowing that it has to be found numerically makes me slightly unhappy I'm glad to find that I wasn't missing anything more fundamental.
  9. Nov 2, 2011 #8
    The erfc(x) is a function which is used exactly like the functions exp(x), ln(x), cos(x) or many others. All these functions have to be found numerically (except for a few particular values of x).
    For example, what is the analytical expressiion of :
    integrate (1/t)*dt from t=1 to t=x ?
    Of course, the answer is ln(x). Then would you say "knowing that it has to be found numerically makes me slightly unhappy" ?

    The only difference between erfc(x) and ln(x) is that one is familiar to you and the other not.

    An almost similar question arised elsewhere about the integration of x^x. The consequence was a funny discussion reported in the paper "The Sophomores Dream Function" :
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