Course Description

Complex analysis is the study of the differential and integral calculus of functions of a complex variable. Complex functions have a rich and tightly constrained structure: for example, in contrast with real functions, a complex function that has one derivative has derivatives of all orders and even a convergent power series. This course develops the theory of complex functions, leading up to Cauchy's theorem and its consequences, including the theory of residues. While the primary viewpoint is calculus, many of the essential insights come from geometry and topology, and we thus will frequently alternate between the analytic and geometric point of view in order to deepen our understanding of the material. 

As we study the question of what continuity, differentiation, integration, and infinite series of complex-valued functions mean, we will see that some of the analysis that we are used to from real-valued functions carries over in the same way, but, more importantly, that complex analysis is also richer and deeper in many ways.  We will also encounter some applications of complex analysis in physics and engineering as well as some beautiful mathematics like the Riemann zeta function and the Fundamental Theorem of Algebra.