Power Series Summary

Complex Analysis · Axiom Academy

Section 8 in one page: how analytic functions become Taylor and Laurent series, where they converge, and what the coefficients reveal. Being analytic at a point is exactly the same as having a convergent power series there — and that forces the function to be infinitely differentiable. A Taylor series (non-negative powers only) represents a function analytic at the center; a Laurent series (negative powers allowed) handles an isolated singularity. The radius of convergence R is the distance from the center to the nearest singularity — finding R locates the singularity. The principal part (the negative-power terms) classifies the singularity: none removable, finitely many pole, infinitely many essential. One coefficient runs the next chapter: the residue , the coefficient of . When f is analytic at z_0 , it equals a power series in (z-z_0) with only non-negative powers . The coefficients are the scaled derivatives — and, equivalently, a contour integral. When to use: the center z_0 is a point of analyticity (no singularity there). To represent a function with an isolated singularity , allow negative powers. The principal part carries all the singular behavior; the analytic part is an ordinary Taylor piece. When to use: there is a singularity at (or inside) the center, so a Taylor series can't reach. Watch out for: the same f has different Laurent series in different annuli around the same center. Core Concept Region of Convergence

This is the written version of the interactive lesson above. See the full Complex Analysis course.