I saw in my lectures notes on Fourier analysis that we call "Poisson's semigroup" the following observations :
Let $\alpha>0$ and $f(x) = \exp(-\alpha \vert x \vert)$. We started by computing $\widehat f$, then using the inversion theorem we deduced the Fourier transform of $x\mapsto \dfrac{1}{1+x^2}$. Moreover we did the Fourier transform of $x\mapsto \dfrac{1}{(1+x^2)^2}$ (using convolution) and finally we computed the Fourier transform of $x\mapsto \dfrac{x}{(1+x^2)^2}$ (using derivatives).
After all of this, I could not figure out why it is called "Poisson's semigroup" and the link with the observations and also what is its use. Did anyone has more precise references about it ?
Thanks in advance !