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Hurwitz theorem

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Let be a sequence of holomorphic functions in a domain that converges uniformly on compact sets in to a function . Then, for any closed rectifiable Jordan curve lying in together with the domain bounded by and not passing through zeros of , it is possible to find a number such that for each of the functions has inside the same number of zeros as inside . Obtained by A. Hurwitz .

References

[1a] A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" Math. Ann. , 46 (1895) pp. 273–284
[1b] A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" , Math. Werke , 2 , Birkhäuser (1933) pp. 533–545
[2] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)


Comments

For another theorem using "nearness of functions" to derive "equality of number of zeros" see Rouché theorem.

How to Cite This Entry:
Hurwitz theorem. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hurwitz_theorem&oldid=14800
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098