Graf's book on hyperfunction theory says (page $36$) that $$\frac1{(x-i0)^n}=\frac{(-1)^{n-1}\pi i}{(n-1)!}\delta^{(n-1)}(x)+\operatorname{fp}\frac1{x^n},$$ while the table of Fourier transforms says that the Fourier transform of $x^n$ is $2\pi i^n\delta^{(n)}(x)$ (here the Fourier transform is...

Working on an algebra of divergent integrals I came to the following relation: If $\tau=\int_0^\infty dx$ then $$\ln (\tau+a)=\int_0^\infty \psi'(x+1)dx + \psi(a+1/2)$$ and this directly gives the following relation: $$\frac1\pi\ln \left(\frac{\tau +\frac{z}{\pi }}{\tau -\frac{z}{\pi }}\right)...