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Difference between revisions of "Rouché theorem"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827102.png" /> be regular analytic functions (cf. [[Analytic function|Analytic function]]) of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827103.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827104.png" />, let a simple closed piecewise-smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827105.png" /> together with the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827106.png" /> bounded by it belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827107.png" /> and let everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827108.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r0827109.png" /> be valid; then in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271010.png" /> the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271011.png" /> has the same number of zeros as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271012.png" />.
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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271014.png" /> be holomorphic mappings (cf. [[Analytic mapping|Analytic mapping]]) of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271015.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271016.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271018.png" />, with isolated zeros, let a smooth surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271019.png" /> homeomorphic to the sphere belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271020.png" /> together with the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271021.png" /> bounded by it and let the following inequality hold on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271022.png" />:
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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 $\Gamma$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271023.png" /></td> </tr></table>
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$$|f(z)|=\sqrt{|f_1(z)|^2+\dotsb+|f_n(z)|^2}>|g(z)|.$$
  
Then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271024.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271025.png" /> the same number of zeros as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271026.png" />.
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Then the mapping $f(z)+g(z)$ has in $G$ the same number of zeros as $f(z)$.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
There is a symmetric form of Rouché's theorem, which says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271028.png" /> are analytic and satisfy the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271032.png" /> have the same number of zeros inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082710/r08271033.png" />. See [[#References|[a2]]]–[[#References|[a3]]] for generalizations of Rouché's theorem in one variable; see [[#References|[a1]]] for the case of several variables.
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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====
 
====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>
 
<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>

Latest revision as of 12:23, 14 February 2020

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 [1]. 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 $f(z)$.

References

[1] E. Rouché, J. Ecole Polytechn. , 21 (1858)
[2] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[3] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)


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 [a2][a3] for generalizations of Rouché's theorem in one variable; see [a1] for the case of several variables.

References

[a1] 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)
[a2] R.B. Burchel, "An introduction to classical complex analysis" , 1 , Acad. Press (1979)
[a3] J.B. Conway, "Functions of one complex variable" , Springer (1978)
How to Cite This Entry:
Rouché theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rouch%C3%A9_theorem&oldid=23514
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article