Cauchy's principal part equation

Question

In the book Many-Body Physics by Coleman, on page 110 there is the following statement:

Using Cauchy's principal part equation, $1/(x-i \delta) = P(1/x) + i \pi \delta(x)$, where $P$ is the principal part.

Here $\delta$ is a number and $\delta(x)$ I presume to be the Dirac delta. I am not sure what this means. I assume it is related to the principal part of a function but, otherwise, I don't know how to obtain this. Help would be appreciated.

Answer 1

This is what’s called the Sokhotski-Plemelj theorem; you can check Wikipedia's article about it.