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Convolution of a Gaussian with itself from the definition!

  1. Jul 19, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the convolution of g(x) = [itex]e^{-πx^{2}}[/itex] with itself from -∞ to ∞ using the definition of convolution, not the Fourier Transform.



    3. The attempt at a solution

    See my attachment. My professor said that you have to use integration by parts, but I keep getting stuck. No matter what I do, I reach the conclusion that the convolution is 0. Is there something that I'm missing here?
     

    Attached Files:

  2. jcsd
  3. Jul 19, 2012 #2

    SammyS

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    Complete the square.

    [itex]\displaystyle -2\pi\left( y^2-xy\right)-\pi x^2 [/itex]
    [itex]\displaystyle =
    -2\pi\left( y^2-xy+\frac{x^2}{4}-\frac{x^2}{4}\right)-\pi x^2 [/itex]

    [itex]\displaystyle =-2\pi\left( y-\frac{x}{2}\right)^2+2\pi\frac{x^2}{4}-\pi x^2 [/itex]

    [itex]\displaystyle =-2\pi\left( y-\frac{x}{2}\right)^2-\pi\frac{x^2}{2} [/itex]​
     
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