Böttcher's equation

Böttcher's equation, named after Lucjan Böttcher, is the functional equation

${\displaystyle F(h(z))=(F(z))^{n}}$

where

• h is a given analytic function with a superattracting fixed point of order n at a, (that is, ${\displaystyle h(z)=a+c(z-a)^{n}+O((z-a)^{n+1})~,}$ in a neighbourhood of a), with n ≥ 2
• F is a sought function.

The logarithm of this functional equation amounts to Schröder's equation.