Functions of a complex variable Complex integration Theorem

Hurwitz's theorem

Builds on Domain in the complex plane Holomorphic function Uniform limits of holomorphic functions are holomorphic Order of a zero

Statement

Let $G$ be a domain in $\mathbb{C}$ , that is, a connected open set. Let $(f_k)$ be a sequence of holomorphic functions on $G$ that converges uniformly on each compact subset of $G$ to a function $f$ . Suppose that $f$ is not identically zero on $G$ , and that $f$ has a zero of order $m$ at a point $z_0 \in G$ . Then for each sufficiently small $\rho > 0$ there is an integer $N$ with the following property. For each $k \ge N$ , the function $f_k$ has exactly $m$ zeros in the disc $|z - z_0| < \rho$ , each counted as many times as its order.

Remarks

The limit $f$ is holomorphic on $G$ by the theorem on uniform limits. A consequence: if no $f_k$ has a zero in $G$ , then $f$ either has no zero in $G$ or is identically zero. A second consequence: if each $f_k$ is injective on $G$ , then $f$ is injective or constant. The radius $\rho$ must be small: $f_n(z) = z - 1 + \frac{1}{n}$ has one zero in the unit disc, but its limit $z - 1$ has none there.

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Complex numbers object
Complex plane concept
Real numbers object
Set bounded above definition
Complex exponential definition
Contour integral definition
Power series in the complex plane definition
Real and imaginary parts definition
Supremum definition
Circle integral definition
Disc of convergence definition
Completeness axiom statement
Existence of square roots of nonnegative reals statement
Square root definition
Modulus of a complex number definition
Complex differentiability definition
Domain in the complex plane definition
Holomorphic function definition
Cauchy's integral formula statement
Zero of a holomorphic function definition
Goursat's theorem statement
Cauchy's integral theorem statement
Power series are holomorphic in the disc of convergence statement
Holomorphic functions are analytic statement
Holomorphic functions on a disc have primitives statement
Morera's theorem statement
Holomorphic functions are infinitely differentiable statement
Order of a zero definition
Uniform limits of holomorphic functions are holomorphic statement
Hurwitz's theorem statement

Tag	`0SSKBQN58M`
URI	`m2://0SSKBQN58M`
Aliases	`hurwitz-theorem-complex-analysis`
Kind	statement, theorem
Writing	W1 stated
MSC	30C15
Status	This copy of the site has no build data. Run `m2 status 0SSKBQN58M`.
Uses	Domain in the complex plane, Holomorphic function, Uniform limits of holomorphic functions are holomorphic, Order of a zero
Same as	Wikidata Q1136043 Wikipedia Hurwitz's theorem (complex analysis)

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