--- title: Power Series description: Power series are infinite series of terms with increasing powers of a variable. Learners will determine intervals and radii of convergence, represent analytic functions as Taylor series, and analyze function behavior near specific points. category: mathematics subcategory: complex-analysis difficulty: beginner, intermediate, advanced url: /subject/power-series --- # Power Series Power series are infinite series of terms with increasing powers of a variable. Learners will determine intervals and radii of convergence, represent analytic functions as Taylor series, and analyze function behavior near specific points. ## Available Resources 2 Books • 3 Courses • 3 Websites ## Websites ### 1. Paul's Online Math Notes - Power Series Lecture notes from Paul Dawkins' Calculus II course at Lamar University that define power series and work through examples of finding the radius and interval of convergence, including checking endpoints, so students can determine exactly where a given series converges. **Difficulty:** Beginner | **Price:** Free **Link:** https://tutorial.math.lamar.edu/classes/calcii/powerseries.aspx **Tags:** power-series, radius-of-convergence, interval-of-convergence, calculus-ii ### 2. 246A, Notes 1: Complex Differentiation **Author:** Terence Tao Graduate lecture notes from Terence Tao's UCLA complex analysis course. Treats formal and convergent power series, radius of convergence via Hadamard's formula, term-by-term differentiation, and why complex differentiability forces analyticity. The single most rigorous free treatment of power series specifically: the post is dominated by power-series material (radius of convergence appears throughout) and sets the structural mental model that holomorphy and local power-series representation are the same thing. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://terrytao.wordpress.com/2016/09/22/246a-notes-1-complex-differentiation/ **Tags:** complex-analysis, power-series, radius-of-convergence, holomorphic-functions, complex-differentiation ### 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 Analysis Steve Brunton's University of Washington crash course on complex analysis for engineers, moving from complex arithmetic and Euler's formula through analytic functions, contour integration, Cauchy's integral formula, and residues, so viewers can evaluate contour integrals that arise in differential equations. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLMrJAkhIeNNQBRslPb7I0yTnES981R8Cg **Tags:** complex-analysis, contour-integration, residue-theorem, cauchy-integral-formula, engineering-math ### 2. 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 ### 3. 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 Analysis (Princeton Lectures in Analysis, Volume II) **Author:** Elias M. Stein, Rami Shakarchi Canonical graduate text. Chapter 1 establishes power series on the complex plane; later chapters derive Taylor and Laurent expansions from Cauchy's integral formula and push into analytic continuation, entire functions, and the zeta function. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0691113858?tag=edmonddante07-20 **Tags:** complex-analysis, power-series, laurent-series, analytic-continuation, cauchy-integral-formula ### 2. Visual Complex Analysis: 25th Anniversary Edition **Author:** Tristan Needham Needham replaces opaque computation with geometric argument across 501 diagrams, treating power series, the disc of convergence, and analytic continuation visually. Answers the 'why' that computational treatments skip — why a power series has a disc rather than an interval of convergence, and why the nearest singularity sets the radius. Paperback edition adds captions explaining each figure plus a Penrose foreword. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0192868926?tag=edmonddante07-20 **Tags:** complex-analysis, power-series, radius-of-convergence, analytic-continuation, geometric-intuition --- *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/power-series