Polzahl

Polzahl, in complex analysis, denotes the order (multiplicity) of a pole of a meromorphic function at a given point. If f is meromorphic in a neighborhood of z0 and f has a pole at z0 of order m ≥ 1, then (z - z0)^m f(z) is holomorphic at z0, and (z - z0)^{m-1} f(z) is not holomorphic at z0. Equivalently, the Laurent expansion of f about z0 has the form f(z) = sum_{k=-m}^{∞} a_k (z - z0)^k with a_{-m} ≠ 0. The pole is simple when m = 1.

Examples: f(z) = 1/(z - z0)^m has a pole of order m at z0. The tangent function tan z has simple poles at z = π/2 + kπ.

Properties: The Polzahl is the multiplicity of the pole and determines the principal part of the Laurent series. For a higher-order pole, the residue a_{-1} may be zero or nonzero. The Polzahl is invariant under multiplication by a holomorphic function that is nonzero at z0. Poles at infinity are studied by considering f(1/w) near w = 0 and defining the order at infinity accordingly.

Applications: Polzahl is used in contour integration, residue theory, and in describing the growth and distribution of values of meromorphic functions.