--- title: Conformal Mapping description: Conformal mapping studies complex functions that preserve angles locally. Learners will understand how to map complicated geometric domains into simpler ones, a technique essential for solving physical problems in fluid dynamics and electrostatics. category: mathematics subcategory: complex-analysis difficulty: beginner, intermediate, advanced url: /subject/conformal-mapping --- # Conformal Mapping Conformal mapping studies complex functions that preserve angles locally. Learners will understand how to map complicated geometric domains into simpler ones, a technique essential for solving physical problems in fluid dynamics and electrostatics. ## Available Resources 2 Books • 2 Courses • 3 Websites • 1 Papers ## 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 ## Papers ### 1. Numerical Conformal Mapping **Author:** Lloyd N. Trefethen Closes the loop between theory and practice: the Riemann mapping theorem is nonconstructive, and this is a current, free, authoritative account by the field's leading numerical analyst of how the maps are computed for real domains. Expository article on how conformal maps are actually computed, covering simply connected domains mapped bijectively to the unit disk, doubly connected domains mapped to circular annuli with a conformal modulus, and canonical slit or multi-disk domains for higher connectivity. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://arxiv.org/abs/2507.14872 **Tags:** conformal-mapping, numerical-analysis, complex-analysis, riemann-mapping-theorem, conformal-modulus ## Books ### 1. Conformal Mapping: Methods and Applications **Author:** Roland Schinzinger, Patricio A. A. Laura Applications-oriented text covering mapping functions, numerical methods, and nonuniform media, then static electric and magnetic fields, transmission lines and waveguides, vibrating membranes and acoustics, plate buckling, elastic stresses, heat conduction, and fluid flow. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/048643236X?tag=edmonddante07-20 **Tags:** conformal-mapping, complex-analysis, electrostatics, fluid-dynamics, heat-conduction, applied-mathematics ### 2. Conformal Mapping **Author:** Zeev Nehari Classic graduate text whose first four chapters build harmonic functions, analytic functions, the complex integral calculus, and families of analytic functions, then develops conformal mapping of simply and multiply connected domains, bridging the theoretical and computational sides. Almost every other book treats conformal mapping as one chapter of complex analysis; this one is the whole book, self-contained from advanced calculus and designed as a one-year graduate course on the subject. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/048661137X?tag=edmonddante07-20 **Tags:** conformal-mapping, complex-analysis, analytic-functions, harmonic-functions, riemann-mapping-theorem ## Websites ### 1. 246A, Notes 5: Conformal Mapping **Author:** Terence Tao Graduate lecture notes treating conformal maps as complex diffeomorphisms, proving the Riemann mapping theorem by Koebe's argument from the Schwarz lemma and Montel's theorem, and covering Green's functions, the Schwarz-Christoffel formula, and components of the uniformisation theorem. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://terrytao.wordpress.com/2016/10/18/246a-notes-5-conformal-mapping/ **Tags:** conformal-mapping, complex-analysis, riemann-mapping-theorem, schwarz-christoffel, uniformization ### 2. Topic 10: Conformal Transformations (MIT 18.04) **Author:** Jeremy Orloff Defines conformal maps as angle-preserving transformations, develops fractional linear (Möbius) transformations and their circle-preserving properties, and applies them to the Dirichlet problem on the unit disk and two-dimensional fluid flow. A lecture notes chapter; readers can move between disk and half-plane to solve boundary-value problems. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/ef46ddd9941227b8700c945bb7475ae9_MIT18_04S18_topic10.pdf **Tags:** conformal-mapping, mobius-transformations, complex-analysis, dirichlet-problem, fluid-flow ### 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 --- *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/conformal-mapping