In mathematical complex analysis, Carathéodory's theorem, proved by , states that if U is a simply connected open subset of the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 C, whose boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

 is a Jordan curve Γ then the Riemann map

f: U → D

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

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

F : Γ → S¹

from Γ to the unit circle

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

 S¹.

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

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

F : cl(U) → cl(D)

from the closure

Closure (mathematics)

In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

 of U to the closed unit disk cl(D) whose restriction

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 to the interior

Interior (topology)

In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....

 is a Riemann map, i.e. it is a bijective holomorphic

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

 conformal map

Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

.

Another standard formulation of Carathéodory's theorem states that for any pair of simply connected

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 open sets U and V bounded by Jordan curves Γ₁ and Γ₂, a conformal map

f : U→ V

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.

A more general version of Carathéodory's theorem is the following. Let

g : D U

be the inverse of the Riemann map, where D ⊂ C is the unit disk, and U ⊂ C is a simply connected domain. Then g extends continuously to

G : cl(D) → cl(U)

if and only if the boundary of U is locally connected.

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

Well-behaved

Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...

.

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

Complement (set theory)

In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

 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

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

by a Jordan curve.