Laurent series

A Laurent series is a power series allowed to run in both directions — not just the usual non-negative powers of (z - z0), but negative powers too. Where a Taylor series can only describe a function that is holomorphic at the center, a Laurent series can describe a function in a ring-shaped region around a point where the function might have a singularity. The negative-power terms are exactly what is needed to capture how a function misbehaves near a bad point.

Formally, on an annulus (a ring) r < |z - z0| < R where f is holomorphic, f has a unique expansion f(z) = sum over all integers n of a_n (z - z0)^n, the sum running from n = -infinity to +infinity. The part with negative n is called the principal part; it encodes the singularity. The coefficients are given by contour integrals a_n = (1/(2pi i)) times the integral of f(z) / (z - z0)^{n+1} dz around a circle in the annulus.

The Laurent expansion is the diagnostic tool that classifies isolated singularities: no negative powers means removable, finitely many means a pole (of order equal to the most negative power present), infinitely many means essential. The coefficient a_{-1} of the (z - z0)^{-1} term is singled out as the residue, because it is the only term whose contour integral around z0 survives — every other power integrates to zero. A caveat: the same function can have DIFFERENT Laurent series in different annuli, so one must specify the ring of convergence.