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Change of variable in integral of product of exponential and gaussian functions

  1. Dec 24, 2012 #1
    I have the integral

    [itex]\int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2}[/itex]

    where [itex]\xi[/itex] is a constant. I would like to transform by some change of variables in the form

    [itex]\int_{-\infty}^{\infty}dx F(x) \int_{-\infty}^{\infty}dy G(y)[/itex]

    the problem is that due to absolute value in the integral one must take in account where x is greater or less than y,

    can someone help me, please?
     
  2. jcsd
  3. Dec 25, 2012 #2

    lurflurf

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    Homework Helper

    First observe that

    [itex]e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2}=e^{-\xi \vert x-y\vert}e^{-(x-y)^2/2}e^{-(x+y)^2/2}[/itex]

    Then you can either change variables such as
    u=(x+y)/sqrt(2)
    v=(x-y)/sqrt(2)
    or break into two regions
    x<y
    x>y
     
  4. Dec 26, 2012 #3
    Hi !

    the clolsed form of the integral involves a special function (erf).
     

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  5. Dec 26, 2012 #4
    Nice trick! Thank you so much!
     
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