Complex Analysis as Catalyst

We see the subject of complex analysis, in both one and several variables, as an excuse to study other areas: real variables, topology, group theory, abstract algebra, partial differential equations, or geometry. The purpose of this paper is to develop that theme, illustrated by several examples. 0 Prefatory Thoughts I am known as a complex analyst, yet I have never had a course in complex variable theory. This lacuna in my education has perhaps had an effect on the way that I view the subject. I see the elegant subject of complex function theory as a stage for the setting of beautiful problems. Once these problems are apprehended or understood, they are often best solved by stripping away the complex variables and bringing in tools from other parts of mathematics. That is the point of view that we promulgate in the present article. We illustrate the point with examples from real analysis, group theory, abstract algebra, partial differential equations, geometry, and other subjects as well. 1 An Example from Topology Topology has become part of the bedrock of modern mathematics. Originally studied as analysis situs, it now permeates all branches of our discipline. We understand that the entire texture of a subject may be influenced by the particular topology that is chosen. The examples in this section illustrate the point. In this paper a domain will be a connected, open set in C or C. Let Ω be a domain. It is a commonplace in complex variable theory to introduce the We are happy to thank the American Institute of Mathematics for its hospitality and support during this work.