Geometric function theory

Study of space and shapes locally given by a convergent power series
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.

In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let f : D → D′ be an orientation preserving homeomorphism between open sets in the plane. If f is continuously differentiable, then it is K quasiconformal if the derivative of f at every point maps circles to ellipses with eccentricity bounded by K.

If K is 0, then the function is conformal.

Analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. The step wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of mathematical singularities. The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology.

Topics in this area include Riemann surfaces for algebraic functions and zeros for algebraic functions.

A Riemann surface, first studied by and named after Bernhard Riemann, is a one dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi valued functions such as the square root and other algebraic functions, or the logarithm.

Topics in this area include "Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations".

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
Although Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function. An example is given below. 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."

The Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is however one of the simplest results capturing the rigidity of holomorphic functions.

Schwarz Lemma. Let D = {z : |z| < 1} be the open unit disk in the complex plane C centered at the origin and let f : D → D be a holomorphic map such that f(0) = 0. Then, |f(z)| ≤ |z| for all z in D and |f′(0)| ≤ 1. Moreover, if |f(z)| = |z| for some non zero z or if |f′(0)| = 1, then f(z) = az for some a in C with |a| (necessarily) equal to 1.

The maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types. Roughly speaking, it says that the maximum of a function in a domain is to be found on the boundary of that domain. Specifically, the strong maximum principle says that if a function achieves its maximum in the interior of the domain, the function is uniformly a constant. The weak maximum principle says that the maximum of the function is to be found on the boundary, but may re occur in the interior as well. Other, even weaker maximum principles exist which merely bound a function in terms of its maximum on the boundary.

the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.