Applications that Depend on Conformal Mapping

Abstract

Part of the utility of conformal mappings is that they can be used to transform a problem on a given domain V to another domain U (see also §§6.1.1). Often we take U to be a standard domain such as the disc $$D = \{ z \in \mathbb{C}:|z| < 1\}$$ (14.1.1.1) or the upper half plane $$U = \{ z \in \mathbb{C}:\operatorname{Im} z > 0\} .$$ (14.1.1.2) Particularly in the study of partial differential equations, it is important to have an explicit conformal mapping between the two domains. In the Appendix to this chapter we give a compendium of conformal mappings of some of the most frequently encountered planar regions.KeywordsHarmonic FunctionDirichlet ProblemConformal MappingNumerical TechniqueHalf PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.