Zero and pole counts weighted by multiplicity and winding number

Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be open, let $f$ be meromorphic on $\mathrm{\Omega }$, and let $\mathrm{\Gamma }$ be admissible for the residue theorem in $\mathrm{\Omega }$. Assume also that $f$ is not identically zero on any connected component of $\mathrm{\Omega }$ and has no zeros on ${\mathrm{\Gamma }}^{\ast }$, so every zero has finite positive order and every index $n(\mathrm{\Gamma },a)$ of Integration over a complex chain and the index of a chain is defined at every zero or pole of $f$.

Here, and in the argument-principle results that use this definition, meromorphic on an open set means meromorphic on every connected component in the sense of Meromorphic functions on a plane domain. The zero and pole sets are the unions of the corresponding componentwise sets.

Whenever only finitely many zeros and poles of $f$ have nonzero index with respect to $\mathrm{\Gamma }$, define the weighted zero count

$Z(f,\mathrm{\Gamma }):={\sum }_{a\in Z(f)}n(\mathrm{\Gamma },a){\mathrm{ord}}_a(f)$

and the weighted pole count

$P(f,\mathrm{\Gamma }):={\sum }_{b\in P(f)}n(\mathrm{\Gamma },b){\mathrm{ord}}_b^{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}(f),$

where ${\mathrm{ord}}_a(f)$ is the zero order from The order of a zero of a holomorphic function and ${\mathrm{ord}}_b^{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}(f)$ is the positive pole order at $b$.

These are finite sums only after a finiteness argument. Under the argument principle hypotheses, that finiteness comes from applying the residue theorem to the logarithmic derivative.