# Syllabus for Math 131, Complex Analysis Harvard University, Fall 2003

## Contact information

Dylan Thurston
dpt@math.harvard.edu
Science Center 420
(617) 495-1938
Office hours Tuesday, 12 noon--1 PM, Thursday, 2:30--3:30PM, and by appointment.
http://math.harvard.edu/~dpt/math113

## Introduction to the course

In this course, we will learn about one-variable complex analysis. This is the study of complex functions of one complex variable. It is simpler and more elegant than the study of functions of two real variables, and in some ways even simpler than the study of functions of one real variable. For instance, a complex function with one continuous first derivative automatically has an infinite number of derivatives, so a whole class of pathological examples in real analysis have no analogues in the complex case. (In fact, more is true: such a function is automatically analytic; that is, it has a power series expansion that converges to the function.)

We will study functions with complex derivatives. One key fact is that, considered as maps from the complex plane to itself, these functions preserve angles (as long as the derivative is not zero). As a result, they are often called conformal maps or functions. (``Conformal'' comes from Latin roots meaning ``same shape''.)

The real strength of complex analysis comes from properties of integrals of complex-analytic functions along paths. In particular, these path integrals remain unchanged when the path is perturbed; this gives a great deal of flexibility when evaluating integrals. A fundamental theorem, the Cauchy residue theorem, reduces the integral of an analytic function along a closed path to counting certain singularities in the region bounded by the path. This is quite powerful, and, indeed, many real integrals can most easily be done using this theorem.

We will wrap up the semester by proving the Riemann mapping theorem: given any two regions in the plane (satisfying some mild assumptions), there is always a conformal bijection between them. (This map is not unique, but we will be able to completely understand the different possibilities.) This has wide-ranging applications, from random walks to drawing maps of the surface of the brain.

## Prerequisites

You should be familiar with real analysis (as taught in Math 23a,b, Math 25a,b, or Math 55a,b), or alternatively have taken traditional calculus (as taught in Math 21a,b) and be familiar with how to write proofs (as taught in Math 101).

In addition, you should have a basic familiarity with complex numbers: you should know how to compute (1+2i)(1-2i).

## Topics

We will follow Bak and Newman's Complex Analysis (required), with additional topics taken from Needham's Visual Complex Analysis (optional) and elsewhere. Topics will include:
• Introduction to complex analysis.
• Complex differentiability: What does it mean for a function to have a complex derivative? An explicit differential equation (the Cauchy-Riemann equations).
• Visualizing complex maps: conformality (preserving angles) and examples.
• Integrals of complex differentiable functions and invariance under deformation of the path of integration.
• One derivative implies infinitely many. Power series expansions and convergence.
• Möbius transformations.
• Harmonic functions.
• Evaluating definite integrals by counting poles enclosed (the Cauchy residue theorem). Applications to real integrals.
• Conformal mappings and the Riemann mapping theorem.