University of Oxford Physics Year 1 - Complex Numbers and Complex Analysis (S01) Complete Lecture Notes & Revision Guide

Comprehensive handwritten lecture notes for the University of Oxford Complex Analysis course. These notes provide a structured and detailed treatment of the material covered throughout the course, including derivations, worked examples, key formulas, diagrams, and explanations designed to support both understanding and revision. Topics covered include: • Complex numbers, Argand diagrams, modulus and argument, Euler's formula, De Moivre's theorem, roots of complex numbers, and polynomial factorisation • Complex exponentials, logarithms, trigonometric and hyperbolic functions, inverse functions, and multivalued complex functions • Functions of a complex variable, analytic functions, differentiability in the complex plane, and power series representations • Cauchy-Riemann equations, harmonic functions, Laplace's equation, and relationships between real and imaginary components of analytic functions • Construction of analytic functions from harmonic functions and applications of harmonic conjugates • Mappings of the complex plane, geometric transformations, stereographic projection, and the Riemann sphere • Conformal mappings, angle-preserving transformations, Jacobians, local behaviour of analytic functions, and criteria for one-to-one mappings • Möbius transformations and mappings between half-planes, strips, wedges, quadrants, circles, and other regions of the complex plane • Applications of complex analysis to electrostatics, potential theory, boundary value problems, and solutions of Laplace's equation in two dimensions • Singularities of complex functions, poles, removable singularities, essential singularities, and classification of singular points • Laurent expansions, residues, residue calculations, and local behaviour near singularities • Complex contour integration, evaluation of real integrals using residues, and powerful methods for solving otherwise difficult integrals The document consists of carefully organised handwritten notes taken during the course and is suitable for lecture review, tutorial preparation, revision, and exam preparation. Ideal for Oxford Physics students and anyone studying complex analysis, mathematical methods, theoretical physics, and advanced calculus.