MAU34205 Topics in complex analysis I

Module Code	MAU34205
Module Title	Topics in complex analysis I
Semester taught	Semester 1
ECTS Credits	5
Module Lecturer	Prof. Dmitri Zaitsev
Module Prerequisites	MAU23204 Introduction to complex analysis

Assessment Details

This module is examined in a 2-hour examination at the end of Semester 1.
Students are assessed based on the exam alone.
Re-assessment, if needed, consists of 100% exam.

Contact Hours

11 weeks of teaching with 3 lectures per week.

Learning Outcomes

On successful completion of this module, students will be able to

Operate with holomorphic functions and branches of multi-valued holomorphic functions.
Give the appropriate definitions, statement and proofs of Cauchy's theorem and its consequences.
Demonstrate the use of the Morera and Riemann extension theorems.
Give examples of power and Laurent series as well as examples of isolated singularities that are removable, poles or essential.

Module Content

Real and complex differentiability, holomorphic functions, branches of multi-valued functions, branches of the logarithmic function and the nth root function, conformal mappings.
Complex integration along piecewise smooth paths, antiderivatives, calculating integrals using antiderivatives, Cauchy's theorem.
Goursat's version for a triangle, for star-shaped regions and their unions, homotopy version.
Elements of homology and the homological version of Cauchy's theorem.
Cauchy's integral formula, power series expansion of holomorphic functions, mean value property, maximum modulus principle, radius and disk of convergence of power series.
Cauchy-Hadamard formula, theorem of Morera, Cauchy's estimates, Liouville's theorem, fundamental theorem of algebra, compact convergence and the Weierstrass theorem.
Order of zeroes, identity principle, Laurent series expansion in a ring, isolated singularities, removable singularities, essential singularities, poles.
Riemann extension theorem, meromorphic functions, Casorate-Weierstrass theorem.
Argument principle, Rouché's theorem, open mapping theorem, the univalence theorem (local injectivity criterion), inverse function theorem.
Spaces of holomorphic functions, seminorms, Montel's theorem, biholomorphic maps between open sets, Riemann mapping theorem.
M¨aut;bius transformations, Riemann sphere (extended complex plane), stereographic projection, rationality of meromorphic functions on the Riemann sphere, automorphisms of the Riemann sphere and the complex plane, Schwarz Lemma.
Automorphisms of the disk, Cayley transform, automorphisms of the upper half-plane, homogeneity of the Riemann sphere, complex plane and disk.