MA4CA2-Complex Analysis II

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:7
Terms in which taught: Spring term module
Pre-requisites: MA2RA2 Real Analysis II and MA2CA1 Complex Analysis I
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA3CA2 Complex Analysis II
Current from: 2023/4

Module Convenor: Dr Santeri Miihkinen
Email: s.j.miihkinen@reading.ac.uk

Type of module:

Summary module description:

This module continues the study of functions of one complex variable.

Aims:

To study more advanced properties of functions of one complex variable and their applications in other areas of mathematics.

Assessable learning outcomes:

By the end of the module, students are expected to be able to:

• Understand the proof of Cauchy’s theorem and integral formula, and use them to obtain further results in complex analysis;


• Use residue calculus to study properties of functions and evaluate integrals;


• Understand the basic geometric properties of Möbius transformations and their importance in complex analysis and applications;


• State and prove a selection of the main theorems;


• Use the main results to treat further problems in complex analysis and its applications.

This module will be assessed to a greater depth than the excluded module MA3CA2.

Additional outcomes:

The use of complex analysis to obtain results in other parts of mathematics.

Outline content:

Cauchy’s theorem and integral formula, and their consequences; series of complex functions; residue calculus; analytic continuation; harmonic functions; Möbius transformations; special functions;Ìýapplications of complex analysis.

More advanced consequences of residue calculus, such as the maximum modulus, argument principle and Rouche’s theorem, will be presented.

Brief description of teaching and learning methods:

Lectures supported by problem sheets.

Summative assessment- Examinations:

2 hours.

Summative assessment- Coursework and in-class tests:

Formative assessment method