Complex Analysis
Steve Brunton
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.
More resources on Power Series
Wolfram MathWorld
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.
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.
Complex Variables with Applications (MIT 18.04)
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.
Complex Analysis (Princeton Lectures in Analysis, Volume II)
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.
Visual Complex Analysis: 25th Anniversary Edition
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.
246A, Notes 1: Complex Differentiation
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.