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.