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Hurwitz's theorem (Theorem)

Define the ball at $ z_0$ of radius $ r$ as $ B(z_0,r) = \{ z \in G : \lvert z-z_0 \rvert < r \}$ and $ D(z_0,r) = \{ z \in G : \lvert z-z_0 \rvert \leq r \}$ is the closed ball at $ z_0$ of radius $ r$.

Theorem 1 (Hurwitz)   Let $ G \subset {\mathbb{C}}$ be a region and suppose the sequence of holomorphic functions $ \{ f_n \}$ converges uniformly on compact subsets of $ G$ to a holomorphic function $ f$. If $ f$ is not identically zero, $ D(z_0,r) \subset G$ and $ f(z) \not= 0$ for $ z$ such that $ \lvert z-z_0 \rvert = r$, then there exists an $ N$ such that for all $ n \geq N$ $ f$ and $ f_n$ have the same number of zeros in $ B(z_0,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 converges to a holomorphic function but that's another theorem altogether), the limit function is not identically zero and furthermore the limit function is not zero on the boundary of some ball, then eventually the functions of the sequence have the same number of zeros inside this ball as does the limit function.

Do note the requirement for $ f$ not being identically zero. For example the sequence $ f_n(z) := \frac{1}{n}$ converges uniformly on compact subsets to $ f(z) := 0$, but $ f_n$ have no zeros anywhere, while $ f$ is identically zero.

Also in general this result holds for bounded convex subsets but it is most useful to state for balls.

An immediate consequence of this theorem is this useful corollary.

Corollary 1   If $ G$ is a region and a sequence of holomorphic functions $ \{ f_n \}$ converges uniformly on compact subsets of $ G$ to a holomorphic function $ f$, and furthermore if $ f_n$ never vanishes (is not zero for any point in $ G$), then $ f$ is either identically zero or also never vanishes.

Bibliography

1
John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.



"Hurwitz's theorem" is owned by jirka.
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See Also: composition algebra over algebaically closed fields, composition algebras over $\mathbb{R}$, composition algebras over finite fields, composition algebras over $\mathbb{Q}$
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Cross-references: point, vanishes, consequence, bounded, eventually, boundary, function, converge, number, compact subsets, converges uniformly, holomorphic functions, sequence, region, radius, ball
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This is version 2 of Hurwitz's theorem, born on 2004-04-12, modified 2005-03-07.
Object id is 5756, canonical name is HurwitzsTheorem.
Accessed 3185 times total.

Classification:
AMS MSC: 30C15 (Functions of a complex variable :: Geometric function theory :: Zeros of polynomials, rational functions, and other analytic functions )
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Some remarks about this article by olegalexandrov on 2005-11-26 22:48:43
Hi there. I have some remarks.

First, the assumption that the function f be not identically zero is not necessary, as it follows from the assumption that f be nonzero on the boundary on the ball.

The assumption that G be a region (open connected set) is not necessary either, since all the action happens in a disk anyway. Ge geing open should be enough.

Thank you for the complex analysis articles. I copied a bunch of them to Wikipedia, including this one.

Oleg
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