Complex Analysis
A branch of mathematical analysis that studies functions of complex numbers. Explore foundational theorems, conformal mappings, residue calculus, and applications across physics, engineering, and number theory.
Cauchy's Integral Theorem
A fundamental theorem in complex analysis stating that the contour integral of a holomorphic function around a closed curve is zero, provided the function is analytic everywhere inside and on the curve.
Introduction to Residue Calculus
A step-by-step guide to evaluating real integrals using complex contour integration. Covers simple poles, higher-order poles, and the residue theorem with worked examples.
Riemann Surfaces in Modern Geometry
Exploring the topological and algebraic structures underlying multi-valued complex functions. Bridges classical analysis with algebraic geometry and string theory compactifications.
Interactive Guide to Conformal Mappings
Visualize angle-preserving transformations of the complex plane. Explore how Möbius transformations, exponential maps, and Joukowski airfoils reshape domains while preserving analyticity.
Liouville's Theorem & Bounded Entire Functions
Proofs, corollaries, and applications of Liouville's theorem. Demonstrates why bounded entire functions must be constant and its role in proving the fundamental theorem of algebra.
Laurent Series & Singularities
Extension of Taylor series to include negative powers. Classifies isolated singularities (removable, poles, essential) and connects to principal parts and residue extraction.