MAU34209 Advanced complex analysis

Assessment Details

• This module is examined in a 2-hour examination at the end of Semester 1.
• Continuous assessment contributes 20% towards the overall mark, consisting of a certain number of homework sets assigned throughout the term.
• Re-assessments if required will consist 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

• Define concepts, prove theorems, and write down examples and counterexamples
• Understand the properties of harmonic functions and special functions such as the Gamma function and the Riemann-Zeta function
• Work with linear fractional transformations and the Riemann sphere
• Construct meromorphic functions with prescribed zeros and poles as well as elementary Riemann surfaces

Module Content

• Singularities, the Casorati-Weierstrass Theorem, elementary value distribution theory, and the Picard Theorems
• Harmonic functions and the Dirichlet Problem
• The stereographic projection and the Riemann sphere
• Linear fractional transformations
• The Riemann Mapping Theorem and the boundary behaviour of the Riemann map
• Divisors, the Mittag-Leffler Theorem, infinite products, and the Weierstrass Theorem on canonical products
• The Gamma Function, the Riemann-Zeta function, and the statement of the Riemann Hypothesis