Residue Calculus (basic)

GRE Math Subject · Axiom Academy

LESSON Residue Calculus (basic) Singularities, Laurent series, residues, and computing integrals A point z₀ is a singularity of f if f is not analytic at z₀ . Removable singularity: Can be "removed" by redefining f at z₀ . Example: f(z) = sin(z)/z at z = 0 Pole of order n: f(z) ~ c/(z-z₀)ⁿ near z₀ . Example: 1/z² has a pole of order 2 at z = 0 Essential singularity: Neither removable nor a pole. Example: e^(1/z) at z = 0 (Picard's theorem: f comes arbitrarily close to any complex value near essential singularities) Simple pole: A pole of order 1. Most common type on subject GRE. Near a singularity z₀ , an analytic (or meromorphic) function can be expanded as a Laurent series : Regular part: Σ aₙ(z-z₀)ⁿ (analytic at z₀ ) Principal part: Σ bₙ/(z-z₀)ⁿ (contains the singularity) Example: f(z) = 1/[z(z-1)] near z = 0: 1/[z(z-1)] = -1/z + (higher order terms) The coefficient of 1/z is -1 (this is the residue!). The residue of f at z₀ , denoted Res(f, z₀) , is the coefficient of 1/(z-z₀) in the Laurent expansion. At z = 0: Res(f, 0) = lim[z→0] z · 1/[z(z-1)] = -1 At z = 1: Res(f, 1) = lim[z→1] (z-1) · 1/[z(z-1)] = 1 The Residue Theorem is the cornerstone of residue calculus: where the sum is over all singularities zₖ inside the contour C (traversed counterclockwise). Why it works: By Cauchy's integral theorem, we can shrink C down to small loops around each singularity. Each small loop integral equals 2πi × residue .

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