Complex Analysis Course

Lesson 1 • Algebra of Complex Numbers

Covers addition, multiplication, modulus, and conjugate operations. Establishes the arithmetic foundation required for all subsequent analysis.

Lesson 2 • Geometry of the Complex Plane

Interprets complex numbers as points and vectors in the plane. Connects algebraic operations to geometric transformations.

Lesson 3 • Topology of the Complex Plane

Defines open sets, neighbourhoods, and connectedness in the complex plane. Provides the topological language used throughout the course.

Lesson 1 • Multi-Valued Functions and Branches

Addresses branch cuts and Riemann surfaces for multi-valued functions. Establishes single-valued analytic branches used in integration.

Lesson 2 • Limits of Complex Functions

Defines limits rigorously using epsilon-delta language adapted to the plane. Links limit behaviour to two-dimensional path dependence.

Lesson 3 • Continuity and Its Properties

Characterises continuous functions and their preservation of topological properties. Connects continuity to uniform continuity on compact sets.

Lesson 1 • Analytic Functions and Harmonic Functions

Establishes that real and imaginary parts of analytic functions are harmonic. Constructs harmonic conjugates via integration.

Lesson 2 • Cauchy-Riemann Equations

Derives the Cauchy-Riemann conditions as necessary and sufficient criteria. Applies them to verify analyticity of standard functions.

Lesson 3 • Complex Derivative Definition

Defines the complex derivative as a limit and contrasts it with real differentiation. Highlights the stronger constraints imposed by complex differentiability.

Lesson 4 • Differentiation Rules and Entire Functions

Extends sum, product, chain, and quotient rules to complex functions. Classifies entire functions and introduces Liouville's theorem informally.

Lesson 1 • Cauchy's Integral Theorem

Proves path independence for analytic functions on simply connected domains. Introduces the deformation of contours principle.

Lesson 2 • Consequences of Cauchy's Theorem

Applies Cauchy's results to prove Liouville's theorem and the fundamental theorem of algebra. Demonstrates the power of analyticity constraints.

Lesson 3 • Cauchy's Integral Formula

Derives the formula expressing function values via contour integrals. Extends to higher-order derivatives and establishes infinite differentiability.

Lesson 1 • Power Series and Radius of Convergence

Analyses power series convergence using ratio and root tests. Establishes that power series define analytic functions inside their disk of convergence.

Lesson 2 • Analytic Continuation

Defines analytic continuation and the identity theorem for analytic functions. Demonstrates how functions extend uniquely beyond their initial domain.

Lesson 3 • Taylor Series of Analytic Functions

Derives Taylor expansions from Cauchy's integral formula. Computes series for standard functions and determines convergence disks.

Lesson 1 • Improper Integrals via Residues

Evaluates improper real integrals using semicircular and rectangular contours. Applies Jordan's lemma to handle exponential decay on arcs.

Lesson 2 • Integrals with Branch Cuts

Handles integrands with algebraic or logarithmic branch cuts using keyhole contours. Extends residue methods to multi-valued functions.

Lesson 3 • Argument Principle and Rouche's Theorem

Uses the argument principle to count zeros and poles inside a contour. Applies Rouché's theorem to locate zeros of perturbed functions.

Lesson 4 • Residues and the Residue Theorem

Defines residues as Laurent coefficients and proves the residue theorem. Connects residues to winding numbers and enclosed singularities.

Lesson 1 • Schwarz-Christoffel Transformation

Derives the Schwarz-Christoffel formula for mapping the upper half-plane to polygons. Applies it to solve Laplace's equation on polygonal domains.

Lesson 2 • Elementary Conformal Maps

Catalogues standard conformal maps including exponential, power, and Joukowski maps. Builds a toolkit for transforming standard geometric regions.

Lesson 3 • Conformal Mapping Theory

Defines conformality via angle and orientation preservation at non-critical points. Links conformality to analyticity and nonzero derivative.

Lesson 4 • Boundary Value Problems via Conformal Maps

Solves Laplace's equation on complex domains by mapping to simpler geometries. Demonstrates the physical relevance of conformal techniques.

Lesson 1 • Picard's Theorems and Value Distribution

States and interprets Picard's little and great theorems on omitted values. Connects to Nevanlinna theory and the distribution of values of entire functions.

Lesson 2 • Meromorphic Functions and Mittag-Leffler

Constructs meromorphic functions with prescribed poles via the Mittag-Leffler theorem. Contrasts with Weierstrass factorisation for zeros.

Lesson 3 • Normal Families and Montel's Theorem

Defines normal families of analytic functions and proves Montel's theorem. Provides the compactness tool used in the proof of the Riemann mapping theorem.

Lesson 4 • The Gamma and Zeta Functions

Analyses the Gamma function as an analytic continuation of the factorial. Introduces the Riemann zeta function and its functional equation.