# Hurwitz theorem

From Encyclopedia of Mathematics

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://www.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