Complex Analysis and Conformal Mapping for Beginners

Complex analysis is a powerful branch of mathematics that simplifies seemingly impossible engineering and physics problems. Understanding complex variables and conformal mappings is essential for anyone looking to excel in advanced engineering, fluid dynamics, and applied mathematics. This text-only course guides you from the absolute basics of complex numbers to the elegant applications of conformal mapping, helping you build strong analytical skills through clear, written explanations and structured practice. What you'll learn: Understand the foundational properties of complex numbers, limits, and analytic functions; Apply the Cauchy-Riemann equations to determine differentiability; Calculate complex integrals using Cauchy's integral formula and residue theory; Solve complex mapping problems by mastering fractional linear transformations and elementary functions; Analyze real-world engineering problems in electrostatics and fluid flow using conformal mappings; Practice step-by-step mathematical derivations with structured written exercises. You will start by establishing a firm grasp of complex arithmetic and limits, gradually progressing to integration, series, and residue calculus. The final sections focus deeply on the geometric properties of analytic functions, specifically focusing on solving conformal mapping problems. This course is designed for students, self-learners, and engineers who have a basic understanding of standard calculus and want a clear, accessible path into complex analysis. No prior knowledge of complex variables is required. Begin your journey into the beautiful world of complex analysis today.