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Gaussian convolution question

  1. Sep 16, 2009 #1
    Greetings,

    I am a computer scientist revisiting integration after a long time. I am stuck with this simple-looking integral that's turning out to be quite painful (to me). I was wondering if one of you could help.

    The goal is to solve the integral

    [tex]
    \int_{0}^{\infty} e^{-(x - t)^2/2 \sigma^2} x^n\ dx .
    [/tex]

    Note that this is the convolution of the Gaussian centered around 0 with the function that equals $x^n$ for $x > 0$, and 0 elsewhere (modulo scaling).

    In particular, I would be interested in seeing any relationship with the integral

    [tex]
    \int_{-\infty}^{\infty} e^{-(x - t)^2/2 \sigma^2} x^n\ dx .
    [/tex]

    which I have worked out.

    Any suggestions?

    Thanks in advance,
    Swar
     
  2. jcsd
  3. Sep 16, 2009 #2

    HallsofIvy

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    If n is odd, that can be done by letting [itex]u= -(x-\mu)^2/(2\sigma^2)[/itex]. If x is even, try integration by parts, letting [itex]u= x^{n-1}[/itex], [itex]dv= xe^{-(x-\mu)^2/(2\sigma^2)}[/itex] to reduce it to n odd.
     
  4. Sep 16, 2009 #3

    mathman

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    As long as t is not 0, the best you can do is express the integral in terms of the error function.
     
  5. Sep 16, 2009 #4
    Regarding error function, that is my guess too, but can you tell me what exactly would need be done?

    I apologize if the question is obvious.

    Best,
    Swar
     
  6. Sep 17, 2009 #5

    mathman

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    What I would do is first let u=x-t. Then xn becomes (u+t)n.
    Expand the polynomial in u and then by succesive itegration by parts, get all the terms to a 0 exponent for u, which will be proportional to erf(t).
     
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