Laurent Series

Complex Analysis · Axiom Academy

When a Taylor series can't reach a singularity, allow negative powers — and the disk of convergence opens into a ring. A Laurent series about z_0 runs over all integer powers — positive and negative: Analytic part (powers ) fills the disk |z-z_0| < R_2 Principal part (negative powers) fills the exterior |z-z_0| > R_1 Each part converges on a different region. Where the two regions overlap — the ring R_1 < |z-z_0| < R_2 — the full series converges. That overlap is the annulus . 2. Same Function, Different Series Take , with poles at z = 0 and z = 1 . About z_0 = 0 those poles carve the plane into two annuli, and f gets a different Laurent series in each. Expand as a geometric series. One negative power: Expand in powers of instead. All negative powers: A geometric series needs its ratio under 1 . Crossing the pole at |z| = 1 flips which expansion converges — so the region you stand in , not the function alone, decides the series. 3. The Principal Part Holds the Residue The principal part — the negative-power terms — is what a Taylor series can never produce. It encodes how f blows up at the singularity, and one coefficient stands out: a_ -1 , the coefficient of , is the residue . Why a_ -1 is special: integrate the series term-by-term around a small loop. Every power (z-z_0)^n integrates to 0 except n = -1 , which contributes . Only the residue survives.

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