3H: Methods in Complex Analysis

Course Aims
The aim of this course is to introduce the theory of analytic functions of one complex variable. The approach taken will be application driven, while setting the background for a rigorous treatment in a later course.

Intended Learning Outcomes of Course
By the end of this course students will be able to:
- state the Cauchy-Riemann equations and derive these from the definition of differentability; use the Cauchy-Riemann equations to determine whether a complex function is analytic on a specified domain; determine whether a function is harmonic and calculate a harmonic conjugate; define and compute with elementary (polynomial, rational, exponential, trigonometric) complex functions; define and compute complex logarithms and powers;
- compute integrals of continuous functions along curves in the complex plane; establish elementary properties of these integrals including the estimation lemma; use the estimation lemma to control integrals of a similar nature to those covered in lectures;
- state Cauchy's theorem for a simple closed path and related results; state and discuss how to prove Taylor's theorem and Cauchy's integral formula for the n-th derivative;
- determine the nature of singularities and compute the residues at poles of suitable meromorphic functions; state the residue theorem, discuss how it can be deduced and use the residue theorem to evaluate real integrals;
- define the notion of a conformal map; give examples of conformal maps between elementary simple domains; establish properties of bilinear maps; compute the image of a line or circle under a bilinear transformation; state the Riemann mapping theorem.