Rouché's theorem

Rouché定理

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

Translation evidence: Literature usage

Meaning in context

If f and g are holomorphic near the closure of a bounded domain with piecewise smooth boundary and |g-f|<|f| on that boundary, then f and g have the same number of interior zeros, with multiplicity.

两个函数在有界区域闭包附近全纯，且边界上 |g-f|<|f| 时，它们在内部的零点个数相同，零点须按重数计数。

Example

On $|z|=2$, $f=z^3$ and $g=z^3+z+1$ satisfy $|g-f|\le3<8$; hence g has three zeros inside.

Usage and distinctions

The strict estimate is needed on the boundary, not throughout the interior; it counts zeros without locating them.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Complex Analysis§17, Proposition 17.6 · Checked 2026-10-14

Translation sources

1. 肇庆学院 —《复变函数》课程教学大纲第六章，学习要求4及第三节，Rouché定理，printed p.10 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.