Math 310: Complex Analysis

Course Content

The course is divided into the following six units. 

Unit 1: Complex Numbers

This unit studies the set of complex numbers, C. We spend time understanding the algebra and topology of C, as well as how to represent numbers in C in various forms (e.g., as points in two-dimensional Euclidean space). We study how to solve basic complex equations (e.g., quadratic equations).

Unit 2: Complex Functions and Mappings

In this unit we explore complex functions. We'll quickly discover that visualizing them requires 4 dimensions. To circumvent this we'll come to think of complex functions as maps from one region of a plane to another region of another plane. We'll then move on to studying the canonical mappings -- linear functions, power functions, etc. -- and conclude with a discussion of the extended complex plane (which formally adds the point "infinity" to C).

Unit 3: Limits and Derivatives of Complex Functions

This unit explores the limits and derivatives of complex functions. This is the unit where we really start to appreciate the richness of complex analysis. We'll discover in this unit that complex differentiability implies a very specific relationship between the real and imaginary parts of a complex function; this relationship is the Cauchy-Riemann equations. We'll then discuss the ramifications of these equations, and their application to fluid dynamics problems.

Unit 4: Integration of Complex Functions

This unit  begins with a study of the complex logarithm and complex trigonometric functions. We then build up the theory of integration of complex functions. Here we'll meet another set of remarkable results, including the Cauchy-Goursat Theorem and Cauchy's integral formulas. Briefly, these use information about complex functions' values on a sufficiently nice closed curve to determine the values of those functions inside those curves. We'll end the unit again discussing the applications of what we will have learned to fluid dynamics.

Unit 5: Complex Sequences and Series

This unit studies complex sequences, complex power series, and the ramifications of complex differentiability and integrability in these contexts. After building up to complex Taylor series, we'll discuss functions with singularities and special power series called Laurent series associated with them. By studying the types of singularities in such series, we'll eventually develop residue theory -- another foundational part of complex analysis -- leading us to Cauchy's Residue Theorem. We'll then explore the myriad applications of this theorem in both complex and real analysis. 

Unit 6: Conformal Mappings

This penultimate unit in the course begins the study of conformal mappings, complex functions that locally preserve angles. Within this category, we'll study fractional linear transformations, in particular, in part for their ability to map domains bijectively onto other domains. (This will lead us to the Riemann Mapping Theorem.) These results have applications to partial differential equations, among other contexts.

Unit 7: The Riemann Hypothesis

This final unit in the course works its way up to the Riemann hypothesis, the most famous unsolved problem in mathematics. In the course of doing so, we study the Riemann zeta function, its connection to prime numbers, and the complex Gamma function. (Along the way we'll also develop a formula for the volume of the n-ball of radius r in n-dimensional Euclidean space.) We'll end the course with the statement of the Riemann hypothesis.