Hurwitz theorem

Let $(f_n(z))$ be a sequence of holomorphic functions in a domain $D \subset \mathbb{C}$ that converges uniformly on compact sets in $D$ to a function $f(z) \not\equiv 0$. Then, for any closed rectifiable Jordan curve $\Gamma$ lying in $D$ together with the domain bounded by $\Gamma$ and not passing through zeros of $f(z)$, it is possible to find a number $N = N(\Gamma)$ such that for $n > N$ each of the functions $f_n(z)$ has inside $\Gamma$ the same number of zeros as $f(z)$ inside $\Gamma$. Obtained by A. Hurwitz .

References

[1a]	A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" Math. Ann. , 46 (1895) pp. 273–284
[1b]	A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" , Math. Werke , 2 , Birkhäuser (1933) pp. 533–545
[2]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)

Comments

For another theorem using "nearness of functions" to derive "equality of number of zeros" see Rouché's theorem.