Laurentreihen

Laurent series (Laurent-Reihe in German) are representations of complex functions by a doubly infinite power series in (z − z0). They generalize Taylor series to include isolated singularities. If a function f is analytic in an annulus r < |z − z0| < R, then it has a Laurent expansion f(z) = sum_{n = −∞}^{∞} a_n (z − z0)^n that converges for all z in that annulus. The coefficients are given by a_n = (1/2πi) ∮ f(ζ) /(ζ − z0)^{n+1} dζ over a positively oriented closed contour within the annulus.

Convergence and singularities: The inner radius r is the distance to the inner boundary; if r > 0, z0 is a singularity. The principal part is sum_{n<0} a_n (z − z0)^n. If only finitely many negative terms occur, z0 is a pole; if infinitely many, z0 is an essential singularity; if all a_n for n<0 vanish, the Laurent series reduces to a Taylor series.

Uniqueness and consequences: The Laurent series is unique for a given annulus, and the radii r and R are determined by the nearest singularities of f. The coefficient a_{−1} is the residue of f at z0, central to the residue theorem in evaluating complex integrals.

Examples and connections: The function f(z) = 1/(z − a) has a Laurent expansion about z0 = a consisting of a single term (z − a)^{−1}. If a function is analytic at z0, its Laurent series reduces to the ordinary Taylor series with no negative powers. Laurent series are fundamental for describing local behavior near singularities and for topics such as analytic continuation and asymptotic analysis.