Hurwitz's theorem (complex analysis)

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 → ∞.{{harvnb|Ahlfors|1966|page=176}}, {{harvnb|Ahlfors|1978|page=178}}

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

:$f\_n(z)\; =\; z-1+\backslash frac\; 1\; n,\; \backslash qquad\; z\; \backslash in\; \backslash mathbb\; C$

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).

Hurwitz's theorem is used in the proof of the Riemann mapping theorem,{{Cite book | last1=Gamelin| first1=Theodore | author1-link=Theodore Gamelin | title=Complex Analysis | publisher=Springer | isbn=978-0387950693| year=2001}} 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.

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 $f\_k\text{'}\; \backslash to\; f\text{'}$ uniformly on the disc, and hence we have another uniform convergence:

: $\backslash frac\{f\_k\text{'}(z)\}\{f\_k(z)\}\; \backslash to\; \backslash frac\{f\text{'}(z)\}\{f(z)\}.$

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

: $m\; =\; \backslash frac\; 1\; \{2\backslash pi\; i\}\backslash int\_\{\backslash vert\; z\; -z\_0\backslash vert\; =\; \backslash rho\}\; \backslash frac\{f\text{'}(z)\}\{f(z)\}\; \backslash ,dz\; =\; \backslash lim\_\{k\backslash to\backslash infty\}\; \backslash frac\; 1\; \{2\backslash pi\; i\}\; \backslash int\_\{\backslash vert\; z\; -z\_0\backslash vert\; =\; \backslash rho\}\; \backslash frac\{f\text{'}\_k(z)\}\{f\_k(z)\}\; \backslash ,\; dz\; =\; \backslash lim\_\{k\backslash to\backslash infty\}\; N\_k$

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

• Rouché's theorem

• {{citation|last=Ahlfors|first= Lars V.|authorlink=Lars Ahlfors|title=Complex analysis. An introduction to the theory of analytic functions of one complex variable|edition=2nd|series= International Series in Pure and Applied Mathematics|publisher= McGraw-Hill|year= 1966}}
• {{citation|last=Ahlfors|first= Lars V.|authorlink=Lars Ahlfors|title=Complex analysis. An introduction to the theory of analytic functions of one complex variable|edition=3rd|series= International Series in Pure and Applied Mathematics|publisher= McGraw-Hill|year= 1978|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.
• {{Springer |title=Hurwitz theorem |id=H/h048160 |first=E.D. |last=Solomentsev}}

Category:Theorems in complex analysis