15. Theorem 2.9.6 - zeros

Let $γ$ be a 1-cycle in $G \subseteq C$ which is the boundary of some open set $A \subseteq G$ . Suppose $f : G \to C$ is meromorphic, and that $γ$ doesn't contain any zeros of $f$ . Then we have:
:?:

\frac{1}{2 π i} \oint_{γ} \frac{f^{'} (z)}{f (z)} d z = # of zeros of f in A with multiplicity

- # poles in A of f counting multiplicity

PROOF:

Use that the logarithmic derivative $\frac{f^{'} (z)}{f (z)}$ is meromorphic, with simple poles at zeros and poles of $f$ .
Use that $γ$ is a boundary and apply residue theorem