<s>[INST] Compute $e^{-x^2} * e^{-x^2}$ How to compute the convolution of $e^{-x^2}$ with itself?
$$e^{-x^2} * e^{-x^2} = \int_{\mathbb R} e^{-(x-y)^2} e^{-y^2}dy = e^{-x^2}\int_{-\infty}^{\infty} e^{2xy - 2y^2} dy$$
I can't solve it. I tried integration by parts so neglecting $e^{-x^2}$:
$$2\int_{-\infty}^{\infty}(2y^2 - xy) e^{2xy - 2y^2}dy$$
please help. thanks.
 [/INST] $$  \left( y - \frac{x}{2} \right)^2 = y^2 - xy +  \frac{x^2}{4}   $$
$$  \left( y - \frac{x}{2} \right)^2 -  \frac{x^2}{4} = y^2 - xy     $$
$$ 2  \left( y - \frac{x}{2} \right)^2 -  \frac{x^2}{2} = 2 y^2 - 2 xy     $$
$$    \frac{x^2}{2}-2  \left( y - \frac{x}{2} \right)^2 =   2 xy- 2 y^2     $$
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