MATH2521 is a Pure Mathematics Level II course about the calculus of complex-valued functions of one complex variable. See the course overview below.

This course has replaced MATH2520 which was previously a 3uoc course.

**Units of credit:** 6

**Assumed knowledge:** MATH1231 or MATH1241 or MATH1251.

**Exclusions:** MATH2621 and MATH2520

**Cycle of offering:** Term 3

**Graduate attributes:** The course will enhance your research, inquiry and analytical thinking abilities.

**More information:** This recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains up-to-date timetabling information.

The higher version of this course, MATH2621 Higher Complex Analysis, is offered yearly in Semester 2.

MATH2521 (alternatively MATH2621) is a compulsory course for Mathematics majors.

If you are currently enrolled in MATH2521, you can log into UNSW Moodle for this course.

#### Course Overview

In first year you studied real-valued functions of a real variable. Real functions vary enormously in how nicely behaved they are in terms of properties like continuity, differentiability and so forth. The library of standard functions that we study, such as rational functions, trigonometric functions, and hyperbolic functions, are all extremely well-behaved; much better even than a 'typical' infinitely differentiable real function.

To understand why, it turns out that one should look at complex functions of a complex variable. Differentiability for these functions, although having an identical looking definition, is a much stronger property than in the real case. The behaviour of complex functions which are differentiable on a reasonable set is very restricted. In a certain sense, all such functions are almost polynomials and this explains many of the properties of our standard real functions.

Studying these complex functions requires us to find new ways of picturing functions. The graphical methods of real variable calculus are of limited use, but geometric reasoning will be just as important.

An important concept throughout the course will be that of a path integral. Here, instead of integrating a function over an interval [a,b], we integrate functions over a curve in the plane. Many of our main theorems will concern how one can evaluate these path integrals with doing any integration! The powerful integral formulas due to Cauchy and Goursat have lots of lovely consequences, including enabling us to do many real integrals for which our real valued methods fail.