Difference between revisions of "Hurwitz theorem"
From Encyclopedia of Mathematics
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+ | Let $(f_n(z))$ be a sequence of [[holomorphic function]]s 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 curve|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==== | ====References==== | ||
− | <table><TR><TD valign="top">[1a]</TD> <TD valign="top"> A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" ''Math. Ann.'' , '''46''' (1895) pp. 273–284</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" , ''Math. Werke'' , '''2''' , Birkhäuser (1933) pp. 533–545</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[1a]</TD> <TD valign="top"> A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" ''Math. Ann.'' , '''46''' (1895) pp. 273–284</TD></TR> | ||
+ | <TR><TD valign="top">[1b]</TD> <TD valign="top"> A. Hurwitz, "Ueber die Bedingungen, unter welchen eine Gleichung nur Würzeln mit negativen reellen Teilen besitzt" , ''Math. Werke'' , '''2''' , Birkhäuser (1933) pp. 533–545</TD></TR> | ||
+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) (Translated from Russian)</TD></TR> | ||
+ | </table> | ||
====Comments==== | ====Comments==== | ||
− | For another theorem using "nearness of functions" to derive "equality of number of zeros" see [[Rouché theorem|Rouché theorem]]. | + | For another theorem using "nearness of functions" to derive "equality of number of zeros" see [[Rouché theorem|Rouché's theorem]]. |
Latest revision as of 18:17, 3 January 2015
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.
How to Cite This Entry:
Hurwitz theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hurwitz_theorem&oldid=14800
Hurwitz theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hurwitz_theorem&oldid=14800
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article