Introduction to Complex Analysis

Understanding complex variables is essential for solving advanced problems in engineering, physics, and applied mathematics. This text-based course demystifies the elegant world of complex analysis, making abstract mathematical concepts intuitive and accessible. You will transition from basic algebra of complex numbers to confidently evaluating intricate integrals and analyzing mapping behaviors. What you'll learn: - Understand the foundational properties of complex numbers and the complex plane - Apply the Cauchy-Riemann equations to determine the analyticity of functions - Evaluate complex contour integrals using Cauchy's Integral Theorem and Formula - Analyze series expansions, identifying Taylor and Laurent series for various functions - Calculate residues and apply the Residue Theorem to evaluate real and complex integrals - Explore conformal mappings and their applications in solving physical boundary-value problems The course begins with foundational definitions, complex algebra, and geometric representations before advancing to limits, continuity, and differentiability. You will then progress through integration theory, singularity analysis, and practical mapping techniques with structured written exercises at every stage. This course is designed for undergraduate students, aspiring engineers, and math enthusiasts who are new to complex variables and have a basic background in single-variable calculus. Start exploring the power of complex functions today.