residue at an isolated singularity

留数

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Translation evidence: Literature usage

Meaning in context

If f is holomorphic on a punctured disk centered at a, its residue at a is the coefficient of (z-a)^(-1) in the Laurent expansion on that disk.

函数在以 a 为中心的去心圆盘全纯时，洛朗展开中 (z-a) 的负一次幂的系数称为 a 处留数。

Example

For f(z)=(2z+1)/(z−2)f(z)=(2z+1)/(z-2), the residue at 22 is 2⋅2+1=52\cdot2+1=5.

Usage and distinctions

A pole can have zero residue: 1/(z−a)21/(z-a)^2 has no inverse-first-power term.

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Definition sources

  1. Michael Taylor — Introduction to Complex Analysis§16, equations (16.1)–(16.5) · Checked 2026-10-14

Translation sources

  1. 肇庆学院 —《复变函数》课程教学大纲第六章，第一节 留数，printed p.10 · Checked 2026-10-14
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