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

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

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, 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 ${\displaystyle f_{k}'\to f'}$ 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.

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

1. Hurwitz's theorem
2. Zeros and poles
3. Adolf Hurwitz
4. Rouché's theorem
5. Gauss–Lucas theorem
6. List of theorems
7. Hurwitz's theorem (composition algebras)
8. Routh–Hurwitz theorem
9. Riemann–Roch theorem
10. Sturm's theorem
11. Riemann–Hurwitz formula
12. Complex number
13. Hurwitz zeta function
14. Routh–Hurwitz stability criterion
15. Riemann mapping theorem
16. Riemann surface
17. Gershgorin circle theorem
18. Bernhard Riemann
19. Carathéodory kernel theorem
20. Quaternion