Geometric function theory
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.
Riemann mapping theorem
Let be a point in a simply-connected region and having at least two boundary points. Then there exists a unique analytic function mapping bijectively into the open unit disk such that and .
It should be noted that while Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function.
In the above figure, consider and as two simply connected regions different from . The Riemann mapping theorem provides the existence of mapping onto the unit disk and existence of mapping onto the unit disk. Thus is a one-to-one mapping of onto . If we can show that , and consequently the composition, is analytic, we then have a conformal mapping of onto , proving "any two simply connected regions different from the whole plane can be mapped conformally onto each other."
Of special interest are those complex functions which are one-to-one. That is, for points , , in a domain , they share a common value, only if they are the same point . A function analytic in a domain is said to be univalent there if it does not take the same value twice for all pairs of distinct points and in , i.e. implies . Alternate terms in common use are schlicht( this is German for plain, simple) and simple. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.