View source for Rouché theorem

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Let $f(z)$ and $g(z)$ be regular analytic functions (cf. [[Analytic function|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 [[#References|[1]]]. It is a corollary of the principle of the argument (cf. [[Argument, principle of the|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|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 $f(z)$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Rouché,   ''J. Ecole Polytechn.'' , '''21'''  (1858)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>



====Comments====
There is a symmetric form of Rouché's theorem, which says that if $f(z)$ and $g(z)$ are analytic and satisfy the inequality $|f(z)+g(z)|<|f(z)|+|g(z)|$ on $\Gamma$, then $f(z)$ and $g(z)$ have the same number of zeros inside $\Gamma$. See [[#References|[a2]]]–[[#References|[a3]]] for generalizations of Rouché's theorem in one variable; see [[#References|[a1]]] for the case of several variables.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.A. Aizenberg,   A.P. Yuzhakov,   "Integral representations and residues in multidimensional complex analysis" , ''Transl. Math. Monogr.'' , '''58''' , Amer. Math. Soc.  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.B. Burchel,   "An introduction to classical complex analysis" , '''1''' , Acad. Press  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.B. Conway,   "Functions of one complex variable" , Springer  (1978)</TD></TR></table>