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 → ∞.^[1]

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

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

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,^[2] 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.^[2]

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 ${\displaystyle f_{k}'\to f'}$ uniformly on the disc, and hence we have another uniform convergence:

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

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

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

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.^[1]

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.
• Solomentsev, E.D. (2001) [1994], "Hurwitz theorem", Encyclopedia of Mathematics, EMS Press

This page was last edited on 26 February 2024, at 15:39