
Module Availability 
Spring 


Assessment Pattern 
Unit(s) of Assessment

Weighting Towards Module Mark( %)


Coursework

25


Written examination (2 hours, unseen)

75


Qualifying Condition(s)
An overall aggregate mark of 40% for the module.





Module Overview 
Complex analysis, traditionally known as the theory of functions of a complex variable, is one of the fundaments of modern mathematics and used in many branches of mathematics and physics. In this module the main results of this theory are presented. 


Prerequisites/Corequisites 
CALCULUS MAT1015 


Module Aims 
This module introduces students to the theory of functions of a complex variable. 


Learning Outcomes 
At the end of the module the student should have a thorough understanding of the theory of complex functions of a complex variable and should be able to apply this knowledge in a variety of contexts. In particular the student should be able to:
* Understand what complex differentiation is.
* Quote, derive and apply the CauchyRiemann equations.
* Study convergence properties of a complex power series
* Perform contour integrations of continuous functions
* Understand and apply Cauchy's theorem and Cauchy's formula
* Derive analyticity and Liouville's theorem from Cauchy's formula
* Apply Taylor's and Laurent's theorems to obtain power series expansions
* Identify and classify singularities of complex functions and find residues of simple
poles.
* Derive and apply the residue theorem to compute real integrals using residue calculus 


Module Content 
Complex functions, complex differentiability; CauchyRiemann equations and harmonic functions; conformal mappings; complex integration: derivation of Cauchy's theorem and Cauchy integration formula, derivation of Liouville theorem and fundamental theorem of algebra; application of Taylor expansion and Laurent expansion; classification of singularities; proof of residue theorem; application to contour integration and evaluation of real integrals. 


Methods of Teaching/Learning 
Teaching is by lectures, tutorials and example classes. Learning takes place through lectures, exercises (example sheets), preparation for tests and background reading.
There will be 3 contact hours for 10 weeks consisting of lectures, tutorials and example classes. 


Selected Texts/Journals 
Recommended:
 I. Stewart and D. Tall, Complex analysis, CUP, (1987).
 M.R. Spiegel, Complex Variables, Schaum's outline, McGraw Hill, (1981).
Further
Reading
:
 J. Marsden and M. Hoffman, Basic Complex Analysis, Freeman & co (1998). 


Last Updated 
September 10 
