Hurwitz's theorem (complex analysis)

Hurwitz's theorem (complex analysis) This article is about a theorem in complex analysis. Für andere Verwendungen, see Hurwitz's theorem.

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.

Inhalt 1 Aussage 2 Bemerkungen 3 Anwendungen 4 Nachweisen 5 Siehe auch 6 References Statement Let {fk} 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), fk 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 zin mathbb {C} } which converges uniformly to f(z) = z − 1. The function f(z) contains no zeroes in D; however, each fn 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 {fn} a sequence of holomorphic functions which converge uniformly on compact subsets of G to a holomorphic function f. If each fn is nonzero everywhere in G, then f is either identically zero or also is nowhere zero. If {fn} 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 {fn} 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| = p. Since fk(z) converges uniformly on the disc we have chosen, we can find N such that |fk(z)| ≥ δ/2 for every k ≥ N and every z on the circle, ensuring that the quotient fk′(z)/fk(z) is well defined for all z on the circle |z − z0| = p. By Weierstrass's theorem we have {Anzeigestil f_{k}'to f'} uniformly on the disc, and hence we have another uniform convergence: {Anzeigestil {frac {f_{k}'(z)}{f_{k}(z)}}zu {frac {f'(z)}{f(z)}}.} Denoting the number of zeros of fk(z) in the disk by Nk, we may apply the argument principle to find {Anzeigestil m={frac {1}{2pi ich}}int _{vert z-z_{0}vert =rho }{frac {f'(z)}{f(z)}},dz=lim _{kto infty }{frac {1}{2pi ich}}int _{vert z-z_{0}vert =rho }{frac {f'_{k}(z)}{f_{k}(z)}},dz=lim _{kto 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 Nk → m as k → ∞. Since the Nk are integer valued, Nk must equal m for large enough k.[1] See also Rouché's theorem References ^ Jump up to: a b Ahlfors 1966, p. 176, Ahlfors 1978, p. 178 ^ Nach oben springen: a b Gamelin, Theodor (2001). Complex Analysis. Springer. ISBN 978-0387950693. Ahlfors, Lars v. (1966), Komplexe Analyse. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics (2und Aufl.), McGraw-Hill Ahlfors, Lars v. (1978), Komplexe Analyse. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics (3Dr. Ed.), McGraw-Hill, ISBN 0070006571 Johannes B. Conway. Funktionen einer komplexen Variablen 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", Enzyklopädie der Mathematik, Kategorien der EMS-Presse: Theoreme in der komplexen Analysis