Prove that $\frac{d}{d x}\log(x+io)=\text{vp}\frac{1}{x}-i\pi \delta(x)$ in the sense of distribution

Question

This is the exercise 4.5.3 of the book

Blanchard, Philippe, and Erwin Brüning. Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics. Vol. 69. Birkhäuser, 2015.

Here $o$ denotes infinitesimal $\epsilon$ with the limiting procedure $\epsilon\rightarrow 0^+$.

Answer 1

When I was asking the question, I did not see the $\frac{d}{d x}$. It is supposed to be a straightforward calculation in terms of the definition of distribution.

\begin{align} \frac{d}{d x}\log(x+io)&=\lim_{\epsilon\rightarrow 0^+}\frac{d}{d x}\log(x+i\epsilon)\\ &=\lim_{\epsilon\rightarrow 0^+}\frac{d}{d x}(\log|x+i\epsilon|+i\arg(x+i\epsilon))\\ &=\lim_{\epsilon\rightarrow 0^+}\frac{x}{x^2+\epsilon^2}-i\frac{\epsilon}{x^2+\epsilon^2}\\ &=\text{vp}\frac{1}{x}-i\pi\delta(x) \end{align}