Complex Analysis — Advanced
University year 3 to graduate levelLearning Goals
- Understand the theory of conformal mappings and their applications
- Learn the statement and significance of the Riemann mapping theorem
- Master the Schwarz–Christoffel transformation for mapping onto polygonal domains
- Understand the concept of analytic continuation and the monodromy theorem
- Study the analytic properties of the gamma function and the zeta function
- Learn the basics of elliptic functions and gain an entry point to modular forms
Prerequisites
- Intermediate-level material (Cauchy's integral theorem, the residue theorem, Laurent series)
- Basics of topological spaces (open sets, connectedness, compactness)
- Fundamentals of real analysis (uniform convergence, continuous functions on compact sets)
Chapters
Chapter 1: Theory of Conformal Mappings
Definition of conformal mappings, conformality of holomorphic functions, and fundamental conformal maps.
Chapter 2: The Riemann Mapping Theorem
Statement and proof outline of the Riemann mapping theorem, and extension to the boundary.
Chapter 3: The Schwarz–Christoffel Transformation
Mappings onto polygonal domains and worked computational examples.
Chapter 4: Analytic Continuation
The concept of analytic continuation, uniqueness, natural boundaries, and the monodromy theorem.
Chapter 5: The Gamma Function and the Zeta Function
Analytic properties of the gamma function and the Riemann zeta function.
Chapter 6: Introduction to Elliptic Functions
Doubly periodic functions, the Weierstrass $\wp$-function, and elliptic integrals.
Chapter 7: Möbius Transformations
Linear fractional transformations, circle-to-circle correspondence, cross-ratio, fixed points, and classification.
Chapter 8: Analytic Automorphisms
Automorphism groups of the unit disk and the upper half-plane, and hyperbolic distance.
Chapter 9: Normal Families and Montel's Theorem
Definition of normal families, Montel's theorem, and its application to the Riemann mapping theorem.
Chapter 10: Approximation Theorems
Runge's theorem and the Mittag-Leffler theorem.
Chapter 11: The Weierstrass Factorization Theorem
Canonical factors, factorization of entire functions, and infinite product representations.
Chapter 12: Picard's Theorem
Picard's little theorem and great theorem, exceptional values, and Bloch's theorem.
Chapter 13: Value Distribution Theory
Nevanlinna theory, the characteristic function, and the first and second main theorems.
Chapter 14: Wirtinger Derivatives
Optimization of real-valued functions with respect to complex variables, and applications to signal processing and machine learning.
Chapter 15: Exercises
A collection of exercises to test your understanding of the advanced material.
Overview
The advanced level covers deep theory and beautiful theorems in complex analysis.
Conformal mappings are angle-preserving maps that give a geometric characterization of holomorphic functions. They find applications in the analysis of many physical phenomena, including fluid dynamics, electrostatics, and heat conduction.
The Riemann mapping theorem is one of the most beautiful theorems in complex analysis. It asserts that every simply connected proper subdomain of the complex plane is conformally equivalent to the unit disk, serving as a fundamental classification result for complex domains.
The Schwarz–Christoffel transformation provides an explicit construction of conformal mappings from the upper half-plane (or the unit disk) onto polygonal domains. It is particularly important in engineering applications.
Analytic continuation is a powerful technique for extending the domain of a function. The monodromy theorem is a key result on the uniqueness of analytic continuation and serves as a gateway to the theory of multivalued functions (Riemann surfaces).
The gamma function $\Gamma(z)$ extends the factorial to complex numbers and is an important function that appears throughout mathematics. The Riemann zeta function $\zeta(s)$ is deeply connected to the distribution of prime numbers, and the Riemann hypothesis — concerning its nontrivial zeros — remains one of the greatest unsolved problems in mathematics.
Elliptic functions are meromorphic functions with two periods, deeply related to elliptic curves and number theory. The Weierstrass $\wp$-function is a fundamental building block of elliptic functions and provides a natural entry point to modular forms.