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Module MA3424: Topics in complex analysis II

Credit weighting (ECTS)
5 credits
Semester/term taught
Hilary term 2017-18
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof. Andreea Nicoara
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
  • 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
Module Prerequisite

MA3423

Assessment Detail

This module will be examined in a 2-hour examination in Trinity term. The final mark is 80% of the exam mark plus 20% continuous assessment consisting of a certain number of homework sets assigned throughout the term. The supplemental examination paper, if required, will determine 100% of the supplemental module mark.