How can I compute $\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$

Question

If $f(x)=\text{arccot}(x)$ for non-negative $x$ and $0$ otherwise, how can I calculate

$$\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$$

for $y\in\mathbb{R}$?

Answer 1

$f\left(x\right)$ is this plot but with $f$ zero for $x < 0$ and $f\left(y-x\right)$ is $f\left(x\right)$ flipped about $x = 0$ and then displaced in the positive $x$ direction by $y$. So, for:

• $y < 0$, there is no overlap, and the integral is zero.
• $y > 0$, the overlap is from 0 to $y$, so you should integrate $\cot^{-1}\left(x\right) \cot^{-1}\left(y-x\right)$ from 0 to $y$.