Singularities Summary

Complex Analysis · Axiom Academy

SUMMARY Singularities and Residues Three kinds of isolated singularity, how to tell them apart, and the one Laurent coefficient that turns an integral into arithmetic. An isolated singularity is exactly one of three types — removable , a pole of order n , or essential — and its Laurent expansion tells you which. The number of negative powers in the Laurent series is the whole story: none → removable, finitely many → pole, infinitely many → essential. The residue is a single number: the coefficient a_ -1 of (z-z_0)^ -1 . For poles you can read it off with a limit formula; at an essential point you take it straight from the series. Behavior near each type is rigid: bounded ⟹ removable (Riemann), ⟹ pole, and wildly dense ⟹ essential (Casorati–Weierstrass, Picard). The payoff is the Residue Theorem: a contour integral equals times the sum of enclosed residues. Core Concept Removable Singularity The "fake" singularity. The Laurent series has no negative powers , so redefining f(z_0) as the limit makes f analytic there. The principal part is empty. Model example: at z=0 , whose limit is 1 . Residue: always 0 (no (z-z_0)^ -1 term). The principal part has finitely many negative powers, the lowest being (z-z_0)^ -n . Then as , and n is the pole's order. Model example: at z=0 is a pole of order 2 . Order shortcut: for , order = (zero-order of h ) − (zero-order of g ). Core Concept Essential Singularity

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