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Module MAU34208: Topics in complex analysis II

Credit weighting (ECTS)

5 credits

Semester/term taught

Hilary term 2019-20

 

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

MAU23204

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. Re-assessments if required will consist of 100% exam.