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| In [[complex analysis]], an area of [[mathematics]], '''Montel's theorem''' refers to one of two [[theorem]]s about [[Family (disambiguation)#Mathematics|families]] of [[holomorphic function]]s. These are named after [[Paul Antoine Aristide Montel|Paul Montel]], and give conditions under which a family of holomorphic functions is [[normal family|normal]].
| | Alyson is the title people use to call me and I believe it sounds fairly good when you say it. For a while I've been in Mississippi but now I'm considering other options. One of the issues she enjoys most is canoeing and she's been doing it for quite a while. Office supervising is where her main income comes from but she's already applied for another one.<br><br>Feel free to visit my weblog; psychic phone readings ([http://ece.modares.ac.ir/mnl/?q=node/805853 linked internet page]) |
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| ==Uniformly bounded families are normal==
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| The first, and simpler, version of the theorem states that a uniformly bounded family of holomorphic functions defined on an [[open set|open]] [[subset]] of the [[complex number]]s is [[normal family|normal]].
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| This theorem has the following formally stronger corollary. Suppose that
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| <math>\mathcal{F}</math> is a family of
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| meromorphic functions on an open set <math>D</math>. If <math>z_0\in D</math> is such that
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| <math>\mathcal{F}</math> is not normal at <math>z_0</math>, and <math>U\subset D</math> is a neighborhood of <math>z_0</math>, then <math>\bigcup_{f\in\mathcal{F}}f(U)</math> is dense
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| in the complex plane.
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| ==Functions omitting two values==
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| The stronger version of Montel's Theorem (occasionally referred to as the [[Fundamental Normality Test]]) states that a family of holomorphic functions, all of which omit the same two values <math>a,b\in\mathbb{C}</math>, is normal.
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| ==Necessity==
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| The conditions in the above theorems are sufficient, but not necessary for normality. Indeed,
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| the family <math>\{z\mapsto z+a: a\in\C\}</math> is normal, but does not omit any complex value.
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| ==Proofs==
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| The first version of Montel's theorem is a direct consequence of [[Marty's Theorem]] (which
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| states that a family is normal if and only if the spherical derivatives are locally bounded)
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| and [[Cauchy's integral formula]].<ref>{{cite book
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| | url = http://books.google.ca/books?id=HwqjxJOLLOoC
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| | title = Progress in Holomorphic Dynamics
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| | author = Hartje Kriete
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| | publisher = CRC Press
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| | year = 1998
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| | pages = 164
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| | accessdate = 2009-03-01
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| }}</ref>
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| This theorem has also been called the Stieltjes–Osgood theorem, after [[Thomas Joannes Stieltjes]] and [[William Fogg Osgood]].<ref>{{cite book
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| | url = http://books.google.ca/books?id=BHc2b0iCoy8C
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| | title = Classical Topics in Complex Function Theory
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| | author = Reinhold Remmert, Leslie Kay
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| | publisher = Springer
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| | year = 1998
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| | pages = 154
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| | accessdate = 2009-03-01
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| }}</ref>
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|
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| The Corollary stated above is deduced as follows. Suppose that all the functions in <math>\mathcal{F}</math> omit the same neighborhood of the point <math>z_0</math>. By postcomposing with the map <math>z\mapsto \frac{1}{z-z_0}</math> we obtain a uniformly bounded family, which is normal by the first version of the theorem.
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| The second version of Montel's theorem can be deduced from the first by using the fact that there exists a holomorphic [[universal covering]] from the unit disk to the twice punctured plane <math>\mathbb{C}\setminus\{a,b\}</math>. (Such a covering is given by the [[elliptic modular function]]).
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| This version of Montel's theorem can be also derived from [[Picard's theorem]],
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| by using [[Bloch's Principle|Zalcman's lemma]].
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| ==Relationship to theorems for entire functions==
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| A heuristic principle known as [[Bloch's Principle]] (made precise by [[Bloch's Principle#Zalcman's lemma|Zalcman's lemma]]) states that properties that imply that an entire function is constant correspond to properties that ensure that a family of holomorphic functions is normal.
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| For example, the first version of Montel's theorem stated above is the analog of [[Liouville's theorem]], while the second version corresponds to [[Picard's theorem]].
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| ==See also==
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| *[[Montel space]]
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| *[[Fundamental normality test]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book | author = John B. Conway | title = Functions of One Complex Variable I | publisher = Springer-Verlag | year = 1978 | isbn=0-387-90328-3 }}
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| *{{springer|title=Montel theorem|id=p/m064890}}
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| *{{cite book | author = J. L. Schiff | title = Normal Families | publisher = Springer-Verlag | year = 1993 | isbn=0-387-97967-0 }}
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| {{PlanetMath attribution|title=Montel's theorem|id=5754}}
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| [[Category:Compactness theorems]]
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| [[Category:Theorems in complex analysis]]
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Alyson is the title people use to call me and I believe it sounds fairly good when you say it. For a while I've been in Mississippi but now I'm considering other options. One of the issues she enjoys most is canoeing and she's been doing it for quite a while. Office supervising is where her main income comes from but she's already applied for another one.
Feel free to visit my weblog; psychic phone readings (linked internet page)