# Math 185: Complex Analysis Lecture 1, Spring 2016

## Course Information

The official course description from the catalog is: Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping.

We will study the additional structure provided by complex differentiation. While some parts of complex analysis will look familiar from real analysis, we will quickly find that in many ways, complex analysis is more rigid. In many ways, the theme of real analysis is that everything that can possibly go wrong can go wrong, and that is why we need to write down rock-solid, rigorous proofs. In contrast, in the complex world, everything that you could possibly wish for (and more) is true. As an example, the additional structure provided by complex differentiation will allow us to compute things like integrals by studying functions just at specific points, without having to do the hard work of finding anti-derivatives. Along the way, we will study other nice properties of complex analytic functions, such as the maximum principle, Liouville's theorem, and conformalness.

This course will assume that you have working knowledge of real analysis from MATH 104 or equivalent. As an advanced upper division course, you are expected to be able to write mathematically rigorous proofs already, and teaching this will not be heavily emphasized as it is in MATH 104. Moreover, many of the concepts will be parallel to those in MATH 104, and it will be expected that you understand things like formal definitions for limits, continuity, and differentiability from the real perspective. In practice, this means that proofs may be omitted or only sketched briefly if they are identical to proofs from real analysis. As such, although some strong students have successfully taken MATH 185 without MATH 104, it is strongly recommended that you complete MATH 104 before taking this course.

## Classroom

This course meets TTh 2:00pm-3:30pm in 458 Evans.

## Miscellaneous

The solutions to the final exam have been posted.

The formula sheet that will be on the final exam is available.

Midterm 2 solutions are available.

Midterm 1 solutions are available.

The course website for the Fall 2014 version of this course is also available.