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Difference between revisions of "Rouché theorem"

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m (moved Rouché theorem to Rouche theorem: ascii title)
m (moved Rouche theorem to Rouché theorem over redirect: accented title)
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Revision as of 07:55, 26 March 2012

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

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 and be holomorphic mappings (cf. Analytic mapping) of a domain of the complex space into , , with isolated zeros, let a smooth surface homeomorphic to the sphere belong to together with the domain bounded by it and let the following inequality hold on :

Then the mapping has in the same number of zeros as .

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 and are analytic and satisfy the inequality on , then and have the same number of zeros inside . 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=22995
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article