<s>[INST] Convolution of a Function I am calculating $f*f$, where $f(x) = \frac{1}{a^2 + x^2}$. So I need to calculate
$$ (f*f)(x) = \int^\infty_{-\infty} \frac{1}{(a^2+y^2)(a^2+(x-y)^2)} dy$$
I guess the answer is $\frac{2\pi}{a(4a^2+x^2)}$. Also, we can assume $a > 0$. I have tried really hard on this problem, and I am at a complete loss. I appreciate your help tremendously!
 [/INST] Things can get easier a lot if you can use Fourier transform. Note that:
$\mathcal{F}(\frac{1}{x^2+a^2})=\frac{1}{a}\sqrt{\frac{\pi}{2}}e^{-a|\omega|}$
$\mathcal{F}(\frac{1}{x^2+a^2}*\frac{1}{x^2+a^2})=(\frac{1}{a}\sqrt{\frac{\pi}{2}}e^{-a|\omega|})(\frac{1}{a}\sqrt{\frac{\pi}{2}}e^{-a|\omega|})=\frac{1}{a^2}\frac{\pi}{2}e^{-2a|\omega|}$
$\frac{1}{x^2+a^2}*\frac{1}{x^2+a^2} = \mathcal{F^{-1}}(\frac{1}{a^2}\frac{\pi}{2}e^{-2a|\omega|})=\frac{\sqrt{2\pi}}{a(4a^2+x^2)}$
Even if you cannot use Fourier transform in your solution may be looking at the proofs related to it help.
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