This week I had to brush up on Green’s functions and I invariably stumbled again on the math of the function

which appears over and over again when one works with the Green function

The first useful thing to know is the *name* of the often used identity

where P.V. stands for principal value of the integral and the function is defined as

The integral form of this useful identity is known as the Sokhatsky–Weierstrass theorem. The integral appearing in the expression above is a Hilbert transform.

The second trick that I found useful deals with the computation of the residue for functions that can be written as a fraction. Say function has a pole in the complex plane at . Then the residue of at is simply given by

This result is useful to compute the residue of expressions like

that often occur in Green’s functions

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