Holomorphic fixed-point index ($\iota (f,z_0)$)

The holomorphic fixed-point index isAt a simple fixed point of multiplier $\lambda \ne 1$, it equals $1/(1-\lambda )$. At infinity, use a local coordinate centred there and the corresponding local expression for the map. For a rational map other than the identity, all fixed points are isolated and their holomorphic fixed-point indices sum to one.

If a quadratic rational map has three distinct fixed points with multipliers ${\lambda }_0,{\lambda }_1,{\lambda }_2$, then ${\lambda }_i{\lambda }_j\ne 1$ for $i\ne j$. Otherwise the two corresponding indices already sum to one, contradicting the nonzero third index.