# Convolution of a Gaussian with itself from the definition!

1. Jul 19, 2012

### GuiltySparks

1. The problem statement, all variables and given/known data

Find the convolution of g(x) = $e^{-πx^{2}}$ 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?

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• ###### gauss.jpg
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2. Jul 19, 2012

### SammyS

Staff Emeritus
Complete the square.

$\displaystyle -2\pi\left( y^2-xy\right)-\pi x^2$
$\displaystyle = -2\pi\left( y^2-xy+\frac{x^2}{4}-\frac{x^2}{4}\right)-\pi x^2$

$\displaystyle =-2\pi\left( y-\frac{x}{2}\right)^2+2\pi\frac{x^2}{4}-\pi x^2$

$\displaystyle =-2\pi\left( y-\frac{x}{2}\right)^2-\pi\frac{x^2}{2}$​