Hurwitz's theorem (complex analysis)
In complex analysis, a field within mathematics, Hurwitz's theorem, named after Adolf Hurwitz, roughly states that, under certain conditions, if a sequence of holomorphic functions converges uniformly to a holomorphic function on compact sets, then after a while those functions and the limit function have the same number of zeros in any open disk.
More precisely, let G be an open set in the complex plane, and consider a sequence of holomorphic functions (f_n) which converges uniformly on compact subsets of G to a holomorphic function f. Let D(z_0,r) be an open disk of center z_0 and radius r which is contained in G together with its boundary. Assume that f(z) has no zeros on the disk boundary. Then, there exists a natural number N such that for all n greater than N the functions f_n and f have the same number of zeros in D(z_0,r).
The requirement that f have no zeros on the disk boundary is necessary. For example, consider the unit disk, and the sequence
- f_n(z) = z-1+\frac{1}{n}
for all z. It converges uniformly to f(z)=z-1 which has no zeros inside of this disk, but each f_n(z) has exactly one zero in the disk, which is 1-1/n.
This result holds more generally for any bounded convex sets but it is most useful to state for disks.
An immediate consequence of this theorem is the following corollary. If G is an open set and a sequence of holomorphic functions (f_n) converges uniformly on compact subsets of G to a holomorphic function f, and furthermore if f_n is not zero at any point in G, then f is either identically zero or also is never zero.
See also
References
- 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