Rouché theorem

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)