In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Koenigs function is a function arising in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
. Introduced in 1884 by the French mathematician
Gabriel Koenigs
Gabriel Xavier Paul Koenigs (17 January 1858 in Toulouse, France – 29 October 1931 in Paris, France) was a French mathematician who worked on analysis and geometry. He was elected as Secretary General of the Executive Committee of the Internat ...
, it gives a canonical representation as dilations of a
univalent holomorphic mapping, or a
semigroup of mappings, of the
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose di ...
in the
complex numbers into itself.
Existence and uniqueness of Koenigs function
Let ''D'' be the
unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose di ...
in the complex numbers. Let be a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
mapping ''D'' into itself, fixing the point 0, with not identically 0 and not an automorphism of ''D'', i.e. a