--- title: Cauchy’s Theorem description: Cauchy's theorem is a fundamental result in complex analysis concerning contour integrals of holomorphic functions. Learners will understand how to evaluate line integrals around closed paths and apply the residue theorem to solve real integrals. category: mathematics subcategory: complex-analysis difficulty: beginner, intermediate, advanced url: /subject/cauchys-theorem --- # Cauchy’s Theorem Cauchy's theorem is a fundamental result in complex analysis concerning contour integrals of holomorphic functions. Learners will understand how to evaluate line integrals around closed paths and apply the residue theorem to solve real integrals. ## Where to start Start with Petra Bonfert-Taylor's Introduction to Complex Analysis, a Wesleyan University course on Coursera that builds from complex numbers to contour integration and Cauchy's theorem. If you only use one resource, make it Jeremy Orloff's free Complex Variables with Applications (MIT 18.04) from MIT OpenCourseWare, covering the theorem, the integral formula and residues. For a rigorous graduate proof, read Terence Tao's Math 246A, Notes 3. ## Available Resources 2 Courses • 2 Websites ## Courses ### 1. Complex Variables with Applications (MIT 18.04) **Author:** Jeremy Orloff Complex analytic functions and their applications: Cauchy's theorem and integral formula, Taylor and Laurent series, residues for evaluating hard integrals, harmonic functions, conformal maps, two-dimensional fluid flow, and Laplace and Fourier transforms. Provides 14 lecture note files plus problem sets and exams with solutions. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/ **Tags:** complex-analysis, analytic-functions, residue-theorem, laurent-series, conformal-mapping ### 2. Introduction to Complex Analysis **Author:** Petra Bonfert-Taylor Wesleyan University course on complex analysis covering complex numbers, analytic functions, conformal mappings, Möbius transformations, contour integration, Cauchy's theorem, power and Laurent series, and residues. Learners finish able to evaluate contour integrals and analyze complex-differentiable functions. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/complex-analysis **Tags:** complex-analysis, analytic-functions, conformal-mapping, contour-integration, residue-theorem ## Youtubes ### 1. Complex integration, Cauchy and residue theorems | Essence of Complex Analysis #6 **Author:** Mathemaniac A 41-minute visual lecture that interprets complex contour integrals through the Pólya vector field as work and flux, then derives Cauchy's theorem, the Cauchy integral formula, and the residue theorem from that picture. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=EyBDtUtyshk **Tags:** complex-analysis, contour-integration, cauchys-theorem, residue-theorem, cauchy-integral-formula ## Websites ### 1. Math 246A, Notes 3: Cauchy's Theorem and Its Consequences **Author:** Terence Tao Graduate lecture notes from Terence Tao's UCLA 246A course. Establishes Cauchy's theorem via Goursat and homotopy arguments, then derives the integral formula, Liouville's theorem, the argument principle, and the residue theorem, with exercises throughout. The rigorous counterweight to the computational courses: Tao states and proves Cauchy's theorem at full generality (winding numbers, homotopy invariance) rather than only for star domains, and shows explicitly how the residue theorem falls out. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://terrytao.wordpress.com/2016/10/02/math-246a-notes-3-cauchys-theorem-and-its-consequences/ **Tags:** complex-analysis, cauchys-theorem, cauchy-integral-formula, residue-theorem, winding-number, lecture-notes ### 2. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/cauchys-theorem