Holomorphic fixed-point index
= Holomorphic fixed-point index
{title2=$\iota(f,z_0)$}
The holomorphic fixed-point index is
$$
\iota(f,z_0)=\operatorname*{Res}_{z=z_0}\frac1{z-f(z)}.
$$
At a simple fixed point of multiplier $\lambda\ne1$, it equals $1/(1-\lambda)$. The indices of all fixed points of a rational map, counted appropriately, sum to one.
= Residue index
{synonym}