--- title: Complex Integration description: Complex integration focuses on integrating complex-valued functions along paths in the complex plane. Learners will master Cauchy's integral theorem and formula, evaluating contour integrals and understanding their applications to real integration. category: mathematics subcategory: complex-analysis difficulty: beginner, intermediate, advanced url: /subject/complex-integration --- # Complex Integration Complex integration focuses on integrating complex-valued functions along paths in the complex plane. Learners will master Cauchy's integral theorem and formula, evaluating contour integrals and understanding their applications to real integration. ## Where to start Start with Complex Variables with Applications (LibreTexts), Jeremy Orloff's free, beginner-level textbook from MIT 18.04, covering contour integrals, Cauchy's theorem and residues. If you only use one resource, make it Complex Variables and Applications by James Ward Brown and Ruel V. Churchill, a paid textbook with a full chapter on residues for real integrals. For a rigorous graduate treatment, move on to Terence Tao's free 246A Complex Analysis Lecture Notes. ## Available Resources 1 Books • 2 Courses • 3 Websites ## Websites ### 1. Complex Variables with Applications (LibreTexts) Jeremy Orloff's free textbook from MIT's 18.04 course, covering complex numbers, analytic functions, contour integrals, Cauchy's theorem, Laurent series, and residues, with applications to Fourier and Laplace transforms. Readers learn to evaluate real integrals and inverse transforms using residue methods. **Difficulty:** Beginner | **Price:** Free **Link:** https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff) **Tags:** complex-analysis, contour-integration, residue-theorem, cauchy-integral-formula, laurent-series, open-textbook ### 2. 246A Complex Analysis Lecture Notes **Author:** Terence Tao Six sets of graduate notes from Tao's UCLA course. Notes 2 through 4 carry the load: contour integration, Goursat's theorem, Cauchy's theorem and integral formula, classification of singularities, Laurent expansion and the residue theorem, with exercises throughout. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://terrytao.wordpress.com/category/teaching/246a-complex-analysis/ **Tags:** complex-analysis, contour-integration, cauchy-theorem, residue-theorem, laurent-series, lecture-notes ### 3. 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 ## 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 ## Books ### 1. Complex Variables and Applications **Author:** James Ward Brown, Ruel V. Churchill The standard applied text since 1948. Chapters four through seven build contour integrals from scratch, then spend a full chapter on residue applications: improper integrals, Jordan's lemma, indented paths, indentation around a branch point, and integration along a branch cut. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0073383171?tag=edmonddante07-20 **Tags:** complex-analysis, contour-integration, residue-theorem, branch-cuts, improper-integrals, textbook ## Youtubes ### 1. Complex Analysis (Berkeley lecture series) **Author:** Richard E. Borcherds Twenty blackboard lectures by a Fields medallist. Eight of them sit squarely on integration: contour integrals, Cauchy's theorem, Cauchy's integral formula, residue calculus, summing series by residues, singularities, and the gamma and zeta functions. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PL8yHsr3EFj537_iYA5QrvwhvMlpkJ1yGN **Tags:** complex-analysis, contour-integration, cauchy-integral-formula, residue-calculus, gamma-function, lectures --- *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/complex-integration