# Rouché theorem

Let $f(z)$ and $g(z)$ be regular analytic functions (cf. Analytic function) of a complex variable $z$ in a domain $D$, let a simple closed piecewise-smooth curve $\Gamma$ together with the domain $G$ bounded by it belong to $D$ and let everywhere on $\Gamma$ the inequality $|f(z)|>|g(z)|$ be valid; then in the domain $G$ the sum $f(z)+g(z)$ has the same number of zeros as $f(z)$.

This theorem was obtained by E. Rouché . It is a corollary of the principle of the argument (cf. Argument, principle of the) and it implies the fundamental theorem of algebra for polynomials.

A generalization of Rouché's theorem for multi-dimensional holomorphic mappings is also valid, for example, in the following form. Let $f(z)=(f_1(z),\dotsc,f_n(z))$ and $g(z)=(g_1(z),\dotsc,g_n(z))$ be holomorphic mappings (cf. Analytic mapping) of a domain $D$ of the complex space $\mathbf C^n$ into $\mathbf C^n$, $n\geq1$, with isolated zeros, let a smooth surface $\Gamma$ homeomorphic to the sphere belong to $D$ together with the domain $G$ bounded by it and let the following inequality hold on $\Gamma$:

$$|f(z)|=\sqrt{|f_1(z)|^2+\dotsb+|f_n(z)|^2}>|g(z)|.$$

Then the mapping $f(z)+g(z)$ has in $G$ the same number of zeros as $f(z)$.

How to Cite This Entry:
Rouché theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rouch%C3%A9_theorem&oldid=44582
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article