This course gives a rigorous treatment of functions of one complex variable focusing on the properties of holomorphic and meromorphic functions, and on conformal mappings.

17 x 1 hr lectures and 6 x 1 hr tutorials in a semester

4H: Further Complex Analysis (MATHS4104)

Assessment

90% Examination, 10% Coursework.

Reassessment

MSc students will have a resit opportunity

It is the default expectation that all courses will offer opportunities for reassessment or deferred assessment. Where it is not possible to offer this in some assessment components, the grade achieved at the first attempt will be counted towards the final course grade, and any exceptions for this course are described below.

[No exceptions]

The course aims to justify the methods developed in the earlier course methods of complex analysis. It will also provide rigorous development of further properties of holomorphic and meromorphic functions and it will end with a study of conformal mappings.

By the end of this course students will be able to:

a) Prove Cauchy's theorem for a simply connected domain, establish standard consequences of Cauchy's theorem and apply these results to problems;

b) Classify the nature of a singularity of a meromorphic function, establish the residue formula, prove the Casorarti-Weierstrass theorem and apply these results to problems;

c) State and prove the principle of the argument, use the principle of the argument to prove Rouche's theorem, use Rouche's theorem to estimate the location of zeros of holomorphic functions, prove the open mapping theorem and maximum modulus principle, apply these results to problems;

d) Define the notion of uniform convergence on compact subsets and equicontinuous and normal families; determine whether given sequences converge uniformly on compact subsets; determine whether given families are normal or equicontinuous; Prove a sequence of holomorphic functions which converges uniformly on compact subsets has a holomorphic limit;

e) Define an infinite product and give product formulas for standard mathematical functions, use infinite products to construct all entire functions with specified zero sets;

f) Define the notion of a conformal mapping and conformal equivalence; construct explicit conformal equivalences between specified complex regions; Use these conformal mappings to solve problems such as the Dirichlet problem in appropriate domains;

g) State and prove Schwarz's lemma, determine all automorphisms of the disk and upper half plane, be able to construct automorphisms with specified behaviour. State and outline the proof of the Riemann mapping theorem.