Convolution of a Gaussian with itself from the definition

[GuiltySparks]

Homework Statement

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

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?

 

[SammyS]

Homework Statement

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

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?

Complete the square.

$\displaystyle -2\pi\left( y^2-xy\right)-\pi x^2$

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

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

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