Definitions
Nearby Words

# Carathéodory's theorem (conformal mapping)

In mathematics, Carathéodory's theorem in complex analysis states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map

f: UD

from U to the unit disk D extends continuously to the boundary, giving a homeomorphism

F : Γ → S1

from Γ to the unit circle S1.

Such a region is called a Jordan domain. Equivalently, this theorem states that for such sets U there is a homeomorphism

F : cl(U) → cl(D)

from the closure of U to the closed unit disk cl(D) whose restriction to the interior is a Riemann map, i.e. it is a bijective holomorphic conformal map.

Another standard formulation of Carathéodory's theorem states that for any pair of simply connected open sets U and V bounded by Jordan curves Γ1 and Γ2, a conformal map

f : UV

extends to a homeomorphism

F: Γ1 → Γ2.

This version can be derived from the one stated above by composing the inverse of one Riemann map with the other.

## Context

Intuitively, Carathéodory's theorem says that compared to general simply connected open sets in the complex plane C, those bounded by Jordan curves are particularly well-behaved.

Carathéodory's theorem is a basic result in the study of boundary behavior of conformal maps, a classical part of complex analysis. In general it is very difficult to decide whether or not the Riemann map from an open set U to the unit disk D extends continuously to the boundary, and how and why it may fail to do so at certain points.

While having a Jordan curve boundary is sufficient for such an extension to exist, it is by no means necessary. For example, the map

f(z) = z2''

from the upper half-plane H to the open set G that is the complement of the positive real axis is holomorphic and conformal, and it extends to a continuous map from the real line R to the positive real axis R+; however, the set G is not bounded by a Jordan curve.

Search another word or see Carathéodory theorem (conformal mapping)on Dictionary | Thesaurus |Spanish