Hurwitz theorem
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hurwitz_theorem&oldid=14800
Hurwitz theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hurwitz_theorem&oldid=14800
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article