Böttcher's equation, named after Lucjan Böttcher, is the functional equation 
formula_1
where 
The logarithm of this functional equation amounts to Schröder's equation.
Solution.
Solution of functional equation is a function in implicit form. 
Lucian Emil Böttcher sketched a proof in 1904 on the existence of solution: an analytic function "F" in a neighborhood of the fixed point "a", such that: 
formula_3
This solution is sometimes called:
The complete proof was published by Joseph Ritt in 1920, who was unaware of the original formulation.
Böttcher's coordinate (the logarithm of the Schröder function) conjugates "h(z)" in a neighbourhood of the fixed point to the function "z""n". An especially important case is when "h(z)" is a polynomial of degree n, and a = ∞ .
Explicit.
One can explicitly compute Böttcher coordinates for: 
Examples.
For the function h and n=2
formula_5
the Böttcher function F is: 
formula_6
Applications.
Böttcher's equation plays a fundamental role in the part of holomorphic dynamics which studies iteration of polynomials of one complex variable. 
Global properties of the Böttcher coordinate were studied by Fatou
 and Douady and Hubbard.