Complex Analysis module (MA32006)

Complex numbers are an extension of the more conventional real numbers that we use in everyday life. They provide a rich structure that helps with finding solutions of equations. Complex numbers are expressed in the form a+bi where an and b are real numbers and i is the imaginary unit. The number i has a fascinating property. When you multiply it by itself you get a negative number, something that never happens with real numbers.

Complex analysis is a branch of maths that studies functions of complex numbers. It extends the concepts of calculus and real analysis to the complex plane. This module explores properties and behaviours of complex functions. This will include differentiation, integration, and series expansions.

Studying complex analysis at university is highly beneficial. It provides a deeper understanding of mathematical concepts and techniques. These are crucial in various fields such as engineering, physics, and computer science. Complex analysis is fundamental in solving problems. These often relate to fluid dynamics, electrical engineering, and quantum mechanics.

What you will learn

In this module, you will:

• learn how to define functions. These include logarithmic, exponential, and trigonometrical functions in terms of complex numbers
• find out how to calculate derivatives and integrals of complex functions
• consider how complex functions can be expanded in series of powers of x and how these series can be used in evaluating integrals.

By the end of this module, you will be able to:

• use advanced results to evaluate complex integrals
• differentiate complex valued functions of a complex variable. These include logarithmic, trigonometric, and exponential functions

Assignments / assessment

• coursework (20%)
• final exam (80%)

Teaching methods / timetable

• two one-hour lectures weekly
  • key points of the week's content will be discussed
  • lecture notes covering the full module content will be available before classes
  • in-class time will be prioritised for interactive discussion
• one hour of tutorials weekly
  • solve problems individually and in groups
  • support with difficulties will be provided by your lecturers and peers