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Approximations to the guassian integral

  1. Oct 20, 2007 #1
    I assume this is standard but I wouldn't know where to look. How do you approximate an integral of the form
    [tex] \int_{x_1}^{x_2}\exp(-x^2)dx[/tex]
    where possibly one of the endpoints is infinite?
     
  2. jcsd
  3. Oct 20, 2007 #2

    Gib Z

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    Homework Helper

    Taking into account that the integrand is an even function, and the well known result that :[tex]\int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}[/tex] It remains only to calculate a integral of the form:

    [tex]\int_0^x e^{-t^2} dt[/tex].

    You may calculate the taylor series and integrate term by term to get a series for that integral: [tex]\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{n! (2n+1)} =\left(x-\frac{x^3}{3}+\frac{x^5}{10}-\frac{x^7}{42}+\frac{x^9}{216}-\ \cdots\right)[/tex],

    Or alternatively, the Error function ( Erf(x) ) is only a constant multiple of that integral, and its values are well tabulated on the internet. http://en.wikipedia.org/wiki/Error_function
     
  4. Oct 21, 2007 #3
    Ah... thanks. That is very useful.
     
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