Complex Analysis — Advanced
Conformal mappings, analytic continuation, special functions, and value distribution
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
- Understand Möbius transformations and the analytic automorphisms of the unit disk and the upper half-plane
- Learn normal families and Montel's theorem, and the approximation theorems of Runge and Mittag-Leffler
- Learn infinite product representations of entire functions via the Weierstrass factorization theorem
- Get to know Picard's theorems and the ideas of value distribution (Nevanlinna) theory
- Learn to optimize real-valued functions of complex variables with Wirtinger derivatives
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.
Reading
Overview
The advanced level covers deep theory and beautiful theorems in complex analysis.
Conformal mappings are angle-preserving maps. A holomorphic function is conformal at each point $z_0$ with $f'(z_0)\neq 0$, while at a point where $f'(z_0)=0$ angles are not preserved (for example, $z^2$ doubles angles at the origin). A global conformal equivalence between domains additionally requires injectivity. Conformal mappings 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 nonempty simply connected proper subdomain of the complex plane is conformally equivalent to the unit disk, serving as a fundamental classification result for complex domains. Its existence proof uses normal families and Montel's theorem, which are treated in Chapter 9.
The Schwarz–Christoffel transformation expresses a conformal mapping from the upper half-plane (or the unit disk) onto a polygonal domain as an integral formula. Apart from three that can be chosen freely, the parameters in the formula (the boundary points that map to the vertices) are determined by nonlinear equations that generally cannot be solved analytically, so they are computed numerically (the parameter problem). It is particularly important in engineering applications.
Analytic continuation is a powerful technique for extending the domain of a function. The monodromy theorem, a key result on the uniqueness of analytic continuation, guarantees that within a simply connected domain the result of continuation does not depend on the path; it 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 that are linearly independent over $\mathbb{R}$ (doubly periodic functions), 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.
Möbius transformations (linear fractional transformations) are the automorphisms of the Riemann sphere and map "circles" (circles and lines) to "circles". With them one can write down all analytic automorphisms of the unit disk and the upper half-plane, which leads to the hyperbolic distance and the Schwarz–Pick theorem.
A normal family is a family of functions in which every sequence has a locally uniformly convergent subsequence — a notion of compactness — and Montel's theorem (a uniformly bounded family of holomorphic functions is normal) is the key to the proof of the Riemann mapping theorem. The approximation theorems are Runge's theorem (approximation of holomorphic functions on compact sets by rational functions) and the Mittag-Leffler theorem (construction of meromorphic functions with prescribed principal parts); the Weierstrass factorization theorem constructs entire functions with prescribed zeros as infinite products.
Picard's theorems state that a nonconstant entire function omits at most one complex value (the little theorem), and that in every punctured neighborhood of an essential singularity a function takes every complex value, with at most one exception, infinitely often (the great theorem). Value distribution theory (Nevanlinna theory) treats quantitatively how values are taken, using the characteristic function and the first and second main theorems.
Wirtinger derivatives differentiate with respect to $z$ and $\bar{z}$ as if they were independent variables, so they also apply to non-holomorphic functions such as real-valued functions of a complex variable. Through the optimization of real-valued functions of complex variables, they are applied in signal processing and machine learning.
Frequently Asked Questions
What advanced complex analysis topics are covered here?
Conformal mappings and the Riemann mapping theorem, the Schwarz–Christoffel transformation, analytic continuation and the monodromy theorem, the gamma and zeta functions, elliptic functions (with an entry point to modular forms), Möbius transformations and analytic automorphisms, normal families and Montel's theorem, approximation theorems (Runge's theorem and the Mittag-Leffler theorem), the Weierstrass factorization theorem, Picard's theorems and value distribution theory (Nevanlinna theory), and Wirtinger derivatives — 14 chapters plus a set of exercises.
What is a Riemann surface and why is it useful?
A Riemann surface is a 1-dimensional complex manifold on which multi-valued functions like $\sqrt{z}$ or $\log z$ become single-valued: a connected Hausdorff space in which a neighborhood of each point is identified with an open subset of the complex plane by a coordinate chart, and the changes of coordinates between overlapping charts are holomorphic. It provides the natural domain for analytic continuation and appears throughout modern mathematics.
How does the Riemann zeta function connect to prime numbers?
For $\mathrm{Re}(s)>1$, Euler's product formula $\zeta(s) = \prod_p (1-p^{-s})^{-1}$ holds and connects the zeta function to all primes. If the Riemann Hypothesis — that all non-trivial zeros satisfy $\mathrm{Re}(s)=1/2$ — is true, then the prime counting function $\pi(x)$ is approximated by the logarithmic integral $\mathrm{Li}(x)=\int_2^x dt/\log t$ as $\pi(x)=\mathrm{Li}(x)+O(\sqrt{x}\log x)$.