请输入您要查询的字词:

 

单词 HurwitzsTheorem
释义

Hurwitz’s theorem


Define the ball at z0 of radius r as B(z0,r)={zG:|z-z0|<r} and D(z0,r)={zG:|z-z0|r} is the closed ballPlanetmathPlanetmath at z0 of radius r.

Theorem (Hurwitz).

Let GC be a region and suppose the sequenceof holomorphic functionsMathworldPlanetmath {fn} converges uniformly on compact subsetsof G to a holomorphic function f. If f is not identically zero,D(z0,r)G and f(z)0 for z such that|z-z0|=r, then there exists an N such that for all nNf and fn have the same number of zeros in B(z0,r).

What this theorem says is that if you have a sequence of holomorphic functions which converge uniformly on compact subsets (such a sequence always convergesPlanetmathPlanetmath to a holomorphic function but that’s another theorem altogether), the function is not identically zero and furthermore the function is notzero on the boundary of some ball,then eventuallythe functions of the sequence have the same number of zeros inside this ball as does the function.

Do note the requirement for f not being identically zero. For example the sequence fn(z):=1n converges uniformly on compact subsets tof(z):=0, but fn have no zeros anywhere, while f is identically zero.

Also in general this result holds for boundedPlanetmathPlanetmathPlanetmathPlanetmath convex subsets (http://planetmath.org/ConvexSet) but it is mostuseful to for balls.

An immediate consequence of this theorem is this useful corollary.

Corollary.

If G is a region and a sequence of holomorphic functions {fn} convergesuniformly on compact subsets of G to a holomorphic function f, and furthermore if fn never vanishes (is not zero for any point in G), thenf is either identically zero or also never vanishes.

References

  • 1 John B. Conway..Springer-Verlag, New York, New York, 1978.
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/5 0:12:07