argument principle

辐角原理

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

Translation evidence: Literature usage

Meaning in context

For a meromorphic f near the closure of a bounded domain with positively oriented piecewise smooth boundary, if f has neither zeros nor poles on the boundary, (2 pi i)^(-1) integral f'/f equals the number of interior zeros minus poles, counted with multiplicity.

函数在有界区域闭包附近亚纯，且在正向分段光滑边界上既无零点又无极点时，f'/f 的边界积分除以 2πi 等于内部零点数减极点数，按重数计算。

Example

For f(z)=(z−1)2/(z+1)f(z)=(z-1)^2/(z+1) inside ∣z∣=2|z|=2, the count is 2−1=12-1=1 and ∫f′/f=2πi\int f'/f=2\pi i.

Usage and distinctions

It counts zeros minus poles, whereas the residue of f itself does not generally count either.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Complex Analysis§17, Propositions 17.1 and 17.9; equations (17.1)–(17.5) · Checked 2026-10-14

Translation sources

  1. 肇庆学院 —《复变函数》课程教学大纲第六章，第三节 辐角原理及其应用，printed p.10 · Checked 2026-10-14
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