A lecture based module, providing an exposition of the basic theory and methods of complex analysis which are fundamental in mathematics and many of its applications.

The course shows how the concepts of differentiation and integration of real functions can be extended to complex functions. Complex functions map complex numbers to complex numbers. For a special subset of these functions it is possible to define a derivative. These differentiable complex functions have particularly nice properties. The real integral between two points x₁ and x₂ on the real axis is generalised to a complex integral along a path between two points z₁ and z₂ in the complex plane. These integrals are called contour integrals. Theorems of Cauchy show how some contour integrals of differentiable complex functions can be evaluated in a beautiful and simple way using methods known as the residue calculus. The residue calculus can be used to evaluate real integrals.

This course is essential for all mathematics students.

• determine whether a complex function is differentiable
• perform integration in the complex plane
• evaluate residues
• apply Cauchy's theorem
• use contour integrals to evaluate real integrals
• understand the proofs of the main theorems of the module

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:
Contour integration and residue calculus.

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 two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

• Revision of complex numbers and elementary functions of a complex variable
• Differentiability and the Cauchy-Riemann equations
• integration in the complex plane
• Cauchy's theorem
• the application of contour integrals to evaluate real integrals
• the argument principle
• Rouche's theorem with applications

Not applicable.

Basic Complex Analysis, Marsden, J. E., & Hoffman, M. J., Freeman