A lecture based module showing some of the ways in which Complex Function Theory developed in the nineteenth century.

The module will show how contour integration may be used to sum certain types of infinite series and to evaluate real integrals involving logarithms or non-integer powers, how Riemann surfaces may be used to represent multivalued functions and how complex functions can be used to map one region of the complex plane to another.

The final topic will be elliptic functions which may be seen as a generalisation of real periodic functions to complex functions which may have two periods.

Prerequisite Modules: MA2003 Complex Analysis

Recommended Modules: MA0221 Analysis III

• use the residue theorem to sum certain types of series
• evaluate real integrals involving logarithms or non-integer powers
• visualise the extended complex plane as the Riemann sphere
• find extended Mobius transformations that map one set of three points onto another
• construct simple conformal mappings from one region to another
• understand the basic properties and construction of elliptic functions

27 – 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills:

An ability to understand contour integration and the geometric representation of functions.

Transferable Skills:

Logical analysis and abstract reasoning.

Formative assessment is carried out by means of regular tutorial exercises. Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of three from four equally weighted questions.

• convergence of series, Laurent's theorem, further applications of contour integration
• the Riemann sphere, Mobius transformations, conformal mappings
• Riemann surfaces and analytic continuation
• elliptic functions

Complex functions: an algebraic and geometric viewpoint, Jones, G. A.,  & Singerman,  D., CUP,  1987

Not applicable.