Residue on all layers of a complex function

Question

I understand how residues work for single-layered functions, but one question type from my assignment has me puzzled.

Say,

$$f(z)={\sqrt{z}\over{(z-z_0)}}$$

has a simple pole at $z=z_0$ and its residue is

$$\text{Res}_{z\to{z_0}} f(z)=\sqrt{z_0}$$

But I'm asked to "find the residues for all layers of a function". What does the second layer "mean" for the residue $\sqrt{z_0}$?

Thanks in advance

Answer 1

Suppose $z_0 \neq 0$ and let $s : U \to \mathbb{C}$ be a fixed branch of $\sqrt{z}$ in a small neighborhood $U$ of $z_0$. Then $f$ has two branches:

$$\frac{s(z)}{z-z_0} \quad \mathrm{and} \quad \frac{-s(z)}{z-z_0}.$$

The first one has the residue at $z=z_0$ equal to $s(z_0)$, the second one has it equal to $-s(z_0)$, that is, the residues are both square roots of $z_0$.