Complex Analysis Fundamentals: Theory and Problem Solving

Understanding functions of a complex variable is essential for mathematics, physics, and modern engineering applications. This text-based course offers an approachable introduction to the fundamental concepts of complex analysis, guiding you from basic arithmetic to advanced integration techniques. You will learn to work confidently with the complex plane, evaluate derivatives of complex functions, apply the Cauchy-Riemann equations, and compute contour integrals using residue theory. What you'll learn: Understand complex arithmetic, polar representations, and mappings on the complex plane; Analyze continuity, differentiability, and analyticity using the Cauchy-Riemann equations; Explore elementary complex functions including exponential, logarithmic, and trigonometric forms; Evaluate path integrals along contours using Cauchy's Integral Theorem and Formula; Calculate Taylor and Laurent series expansions for complex functions; Apply residue calculus to solve challenging real and complex integrals. The course begins with core terminology and algebraic foundations before advancing step by step through differentiation, integration, and series representations. Designed for learners with a basic background in single-variable calculus, this course requires no prior knowledge of complex analysis. Start reading today to deepen your mathematical problem-solving skills.