Complex Analysis II: Further Topics

Abstract

There are some extremely important concepts in complex analysis which we did not cover in Chapter 10, and which ultimately lead up to several other areas of mathematics. First of all, quite a bit more can be said about conformal maps. Under very general conditions, one open subset of $\mathbb{C}$ can be mapped holomorphically bijectively onto another. We prove one such result, the famous Riemann Mapping Theorem. In many situations, such maps can even be written down explicitly. Those are the Schwartz-Christoffel formulas, which have applications in cartography, as the basic condition on mappings in cartography is to be conformal (since distortion of distances in a topographical map is generally considered more allowable than distortion of angles). Yet, the Schwarz-Christoffel formulas also lead to elliptic integrals, which are “inverse” to elliptic functions (see for example [11]).