Complex Analysis

Objectives:

The course aims to familiarize the student with the most important properties and results concerning holomorphic functions.

Content:

This course explores the fundamental properties of functions of a complex variable. A function is said to be holomorphic if it has a complex derivative throughout its domain, leading to several remarkable properties. For example, a holomorphic function is entirely determined by its values on any arbitrarily small subdomain, and it can be expanded into a Taylor series around any point.

We will use these properties to compute integrals using residue calculus and explore how domains can be deformed in a way that preserves angles - known as conformal mapping. Additionally, we will study how root and logarithm functions for complex numbers are multivalued functions and how branch cuts can be introduced to make them single-valued.

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student

• should be able to determine if a function is holomorphic using the Cauchy-Riemann equations.
• should be able to understand Cauchy's integral formula for holomorphic functions and its relation to series expansion.
• should be able to use Taylor and Laurent series expansions for holomorphic and meromorphic functions, including applications in solving integrals using residue calculus.
• should be able to understand key functions such as the exponential function, logarithmic functions, roots, and trigonometric functions, including their multivalued nature and the concept of branch cuts.
• should be able to apply holomorphic functions in contexts such as harmonic functions and Dirichlet's problem.
• should be able to recognize central results such as the Fundamental Theorem of Algebra, Liouville's theorem for entire functions, and the Riemann mapping theorem.

Skills:

The student:

• masters fundamental techniques in complex analysis and complex integration, and knows how to apply these in both theoretical and applied problems.
• can argue mathematically and present proofs and reasoning.

General Competence:

The student:

• is able to work independently.
• is able to express themselves in a precise and scientific manner.
• is able to use logical thinking and determine whether simple mathematical arguments are correct.