Hurwitz's theorem (complex analysis)

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

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:

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:

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

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.