Hurwitz's theorem (complex analysis)

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.

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

[math]\displaystyle{ f_n(z) = z-1+\frac 1 n, \qquad z \in \mathbb C }[/math]

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, 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.

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 Weierstrass's theorem we have [math]\displaystyle{ f_k' \to f' }[/math] uniformly on the disc, and hence we have another uniform convergence:

[math]\displaystyle{ \frac{f_k'(z)}{f_k(z)} \to \frac{f'(z)}{f(z)}. }[/math]

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

[math]\displaystyle{ 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 }[/math]

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 the N_k are integer valued, N_k must equal m for large enough k.

This proof contains a gap towards the end. Indeed, one should still argue why the functions [math]\displaystyle{ f_k }[/math] do not have any other zeroes or poles inside the small disc.

See also

• Rouché's theorem

References

• Ahlfors, Lars V. (1966), Complex analysis. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics (2nd ed.), McGraw-Hill 
• Ahlfors, Lars V. (1978), Complex analysis. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics (3rd ed.), McGraw-Hill, ISBN 0070006571 
• 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.
• Hazewinkel, Michiel, ed. (2001), "Hurwitz theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=H/h048160