LaurentReihe

LaurentReihe, or Laurent series, is a fundamental concept in complex analysis describing the local behavior of an analytic function near an isolated singularity. If a function f is analytic on a punctured neighborhood of z0, then there exists an annulus r < |z - z0| < R (with 0 ≤ r < R ≤ ∞) and a doubly infinite series f(z) = sum_{n=-∞}^{∞} a_n (z - z0)^n that converges to f for all z in that annulus. If r = 0 and R = ∞, the series reduces to a Taylor series.

The coefficients a_n are uniquely determined by contour integrals: a_n = (1/2π i) ∮ f(ζ)/(ζ - z0)^{n+1} dζ, where the contour lies in the annulus and circles z0 once counterclockwise. The part with negative n, called the principal part, encodes the nature of the singularity at z0. If all negative coefficients vanish, z0 is a removable singularity. If only finitely many negative coefficients are nonzero, z0 is a pole of order m, with a_{-m} ≠ 0. If infinitely many negatives occur, z0 is an essential singularity.

Laurent series are unique for a given annulus, and they facilitate many tools such as the residue theorem and contour integration. They also provide analytic continuation and asymptotic information. A Laurent series around z0 is a local representation; by selecting different centers, one obtains expansions valid in corresponding annuli.