pole of a complex function

极点

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

An isolated singularity a of f is a pole of order m>=1 if f(z)=g(z)/(z-a)^m near a with g holomorphic and g(a) nonzero.

孤立奇点 a 附近若 f 可写为全纯函数 g 除以 (z-a)^m，且 g(a) 不为零、m 为正整数，则 a 是 m 阶极点。

Example

f(z)=(z+3)/(z−1)2f(z)=(z+3)/(z-1)^2 has a pole of order 22 at 11 since the numerator there is 44.

Usage and distinctions

At a pole the Laurent principal part is finite and nonempty; an essential singularity has infinitely many negative terms.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Complex Analysis§11, Proposition 11.2 and equation (11.8); §12, Proposition 12.2 · Checked 2026-10-14

Translation sources

  1. 肇庆学院 —《复变函数》课程教学大纲第五章，第二节 解析函数的孤立奇点，极点，printed p.9 · Checked 2026-10-14
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