Hurwitz's theorem (complex analysis)

This article is about a theorem in complex analysis. For other uses, see Hurwitz's theorem.

In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit. The theorem is named after Adolf Hurwitz.

Theorem statement

Let {f_k} be a sequence of holomorphic functions on a connected open set G that converge uniformly on compact subsets of G to a holomorphic function f which is not constantly zero on G. If f has a zero of order m at z₀ then for every small enough ρ > 0 and for sufficiently large k ∈ N (depending on ρ), f_k has precisely m zeroes in the disk defined by |z−z₀| < ρ, including multiplicity. Furthermore, these zeroes converge to z₀ as k → ∞.

Remarks

The theorem does not guarantee that the result will hold for arbitrary disks. Indeed, if one chooses a disk such that f has zeroes on its boundary, the theorem fails. An explicit example is to consider the unit disk D and the sequence defined by

f_n(z) = z-1+\frac{1}{n}, \qquad z \in \mathbb C

which converges uniformly to f(z) = z−1. The function f(z) contains no zeroes in D; however, each f_n has exactly one zero in the disk corresponding to the real value 1−(1/n).

Applications

Hurwitz's theorem is used in the proof of the Riemann Mapping Theorem,^[1] and also has the following two corollaries as an immediate consequence:

Let G be a connected, open set and {f_n} a sequence of holomorphic functions which converge uniformly on compact subsets of G to a holomorphic function f. If each f_n is nonzero everywhere in G, then f is either identically zero or also is nowhere zero.
If {f_n} is a sequence of univalent functions on a connected open set G that converge uniformly on compact subsets of G to a holomorphic function f, then either f is univalent or constant.^[1]

Proof

Let f be an analytic function on an open subset of the complex plane with a zero of order m at z₀, and suppose that {f_n} is a sequence of functions converging uniformly on compact subsets to f. Fix some ρ > 0 such that f(z) ≠ 0 in 0 < |z−z₀| ≤ ρ. Choose δ such that |f(z)| > δ for z on the circle |z−z₀| = ρ. Since f_k(z) converges uniformly on the disc we have chosen, we can find N such that |f_k(z)| ≥ δ/2 for every k ≥ N and every z on the circle, ensuring that the quotient f_k′(z)/f_k(z) is well defined for all z on the circle |z−z₀| = ρ. By Morera's theorem we have a uniform convergence:

\frac{f_{k}'(z)}{f_{k}(z)} \to \frac{f'(z)}{f(z)}.

Denoting the number of zeros of f_k(z) in the disk by N_k, we may apply the argument principle to find

m = \frac{1}{2\pi i}\int_{\vert z -z_{0}\vert = \rho} \frac{f'(z)}{f(z)} \,dz = \lim_{k\to\infty}\frac{1}{2\pi i}\int_{\vert z -z_{0}\vert = \rho} \frac{f'_{k}(z)}{f_{k}(z)} \,dz = \lim_{k\to\infty}N_{k}

In the above step, we were able to interchange the integral and the limit because of the uniform convergence of the integrand. We have shown that N_k → m as k → ∞. Since N_k are integer valued, N_k must equal m for large enough k.

References

1 2 Gamelin, Theodore (2001). Complex Analysis. Springer. ISBN 978-0387950693.

John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
E. C. Titchmarsh, The Theory of Functions, second edition (Oxford University Press, 1939; reprinted 1985), p. 119.
Solomentsev, E.D. (2001), "Hurwitz theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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