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

$f_n(z) = z-1%2B\frac{1}{n}$

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.

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), "Hurwitz theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=H/h048160

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Hurwitz's theorem (complex analysis)

See also

References