Conformal Mapping Of Multiply Connected Domains Research Articles

Technical Report: Computation on the Extended Complex Plane and Conformal Mapping of Multiply-connected Domains

We introduce a system for computation on the extended complex plane based on the Type-Two Effectivity approach to computable analysis. Included are computations on meromorphic functions, open sets, and closed sets. Applications to Möbius transformations, boundaries of multiply connected domains, and conformal mapping of multiply connected domains are considered.

An Orthonormalization Method for the Approximate Conformal Mapping of Multiply-Connected Domains

An Orthonormalization Method for the Approximate Conformal Mapping of Multiply-Connected Domains

On conformal mapping of multiply-connected domains

On conformal mapping of multiply-connected domains

The Sharp Form of the Principle of Hyperbolic Measure

Schwarz's Lemma and its generalizations, the Lindel6f Principle and the Principle of Hyperbolic Measure, have provided an important tool for the study of the theory of functions. From a knowledge of the range of values of an analytic function F(z) in a given domain one is able, with their aid, to obtain estimates on the range of values which the function and its derivative can assume in any subdomain of the given domain. However, these estimates are not in general sharp when the domain of definition of the analytic function F(z) is multiply-connected. Thus considerable interest has been aroused in the problem of finding sharp estimates corresponding to the estimates of these fundamental principles. The work of Teichmiiller [13], Grunsky [6] and Ahlfors [1] has led to a method which yields the precise estimates and the associated extremal functions in the case where bounds on the modulus of the function F(z) are given. This method consists in applying Green's formula to the harmonic function log I F(z) 1. However, if the values of F(z) are assumed to lie in a multiply-connected domain, rather than in a circle corresponding to an upper bound on i F(z) 1, the above method for obtaining sharp estimates breaks down. The difficulty lies in the fact that the Green's function of a multiply-connected domain has critical points. In this paper we attack the general problem of obtaining estimates on F(z) when its domain of definition and range of values are both known multiplyconnected domains. We utilize the technique of Teichmfiller, Grunsky and Ahlfors, while at the same time we are able to avoid the difficulties inherent in the general problem by using the variational method developed by Schiffer [11, 12]. Our results will underline the fact that Schiffer's method is by no means restricted to the study of schlicht or p-valued functions. Our approach will also show that many extremal problems in the conformal mapping of multiply-connected domains can be treated in almost the same manner as one treats similar problems for simply-connected domains.

Mapping by analytic functions. I. Conformal mapping of multiply-connected domains

1. Methods employed. In the theory of analytic functions of one complex variable it is possible to obtain many results of far-reaching importance by means of relatively few general theorems, for example, Riemann's mapping theorem and the lemma of Schwarz. The existence of a non-Euclidean metric which is invariant with respect to conformal mapping follows from Riemann's mapping theorem and is in many instances equivalent to it. In the case of a simply-connected domain B with more than one boundary point, the distance (in this sense) between two points a and b of B is defined as the hyperbolic distance between the images of a and b in the unit circle. By means of this metric it is possible to give to Schwarz's lemma a new and useful formulation, the lemma of Schwarz-Pick. Completely avoiding Riemann's theorem, Bergman succeeded in showing that an invariant Hermitian metric can be derived from a quite different approach and can be generalized to the case of several complex variables. This is important, since Poincare has shown that it is not in general possible to map one domain in 2n-dimensional space on another by means of n analytic functions of n complex variables (pseudo-conformal mapping), so that an immediate generalization of classical function theoretic methods is not possible. Further, in the case of mappings into schlicht domains, Bergman has obtained results which include the lemma of Schwarz-Pick but which can also be applied to the case of several complex variables. However, if those methods originally employed to provide results in pseudo-conformal mapping are specialized to the theory of analytic functions of one complex variable, they do more than merely provide known results: they constitute methods for the investigation of many questions in conformal mapping of multiply-connected domains. Further, theorems that we may obtain in the case of one variable by these methods suggest analogous theorems for pseudo-conformal mapping. It is this dual role which makes both the theorems obtained and the methods employed take on added importance (1).