en>Wtshymanski: pompous "naturally"; lower case, ce; lower case in headings

2014-11-23T07:13:20Z

pompous "naturally"; lower case, ce; lower case in headings

en>Faizhaider: Disambiguated: Kirchhoff's Law → Kirchhoff's circuit laws

2014-02-05T18:53:21Z

Disambiguated: Kirchhoff's Law → Kirchhoff's circuit laws

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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])

	~~==Uniformly bounded families are normal==~~
	~~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]].~~

	~~This theorem has~~ the ~~following formally stronger corollary. Suppose that~~
	~~<math>\mathcal{F}</math>~~ is ~~a family of~~
	~~meromorphic functions on an open set <math>D</math>. If <math>z_0\in D</math> is such that~~
	~~<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~~
	~~in the complex plane.~~

	~~==Functions omitting two values==~~
	~~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.~~

	~~==Necessity==~~
	~~The conditions in the above theorems are sufficient,~~ but ~~not necessary~~ for ~~normality~~. ~~Indeed,~~
	~~the family~~ <~~math~~>~~\{z\mapsto z+a: a\in\C\}~~<~~/math~~> ~~is normal, but does not omit any complex value.~~

	~~==Proofs==~~
	~~The first version of Montel's theorem is a direct consequence of [[Marty's Theorem]]~~ (~~which~~
	~~states that a family is normal if and only if the spherical derivatives are locally bounded)~~
	~~and [~~[~~Cauchy's integral formula]].<ref>{{cite book~~
	~~\| url =~~ http://~~books~~.~~google~~.~~ca/books?id=HwqjxJOLLOoC~~
	~~\| title = Progress in Holomorphic Dynamics~~
	~~\| author = Hartje Kriete~~
	~~\| publisher = CRC Press~~
	~~\| year = 1998~~
	~~\| pages = 164~~
	~~\| accessdate = 2009-03-01~~
	~~}}</ref>~~

	~~This theorem has also been called the Stieltjes–Osgood theorem, after [[Thomas Joannes Stieltjes]] and [[William Fogg Osgood]]~~.~~<ref>{{cite book~~
	~~\| url = http:~~//~~books.google.ca/books~~?~~id=BHc2b0iCoy8C~~
	~~\| title = Classical Topics in Complex Function Theory~~
	~~\| author = Reinhold Remmert, Leslie Kay~~
	~~\| publisher = Springer~~
	~~\| year = 1998~~
	~~\| pages~~ = ~~154~~
	~~\| accessdate = 2009-03-01~~
	~~}}<~~/~~ref>~~

	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.

	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]]).

	~~This version of Montel's theorem can be also derived from [[Picard's theorem]],~~
	~~by using [[Bloch's Principle\|Zalcman's lemma]].~~

	~~==Relationship to theorems for entire functions==~~
	~~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.~~

	~~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]].~~

	~~==See also==~~
	*[[Montel space]]
	*[[Fundamental normality test]]

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*{{cite book \| author = John B. Conway \| title = Functions of One Complex Variable I \| publisher = Springer-Verlag \| year = 1978 \| isbn=0-387-90328-3 }}
	*{{springer\|title=Montel theorem\|id=p/m064890}}
	*{{cite book \| author = J. L. Schiff \| title = Normal Families \| publisher = Springer-Verlag \| year = 1993 \| isbn=0-387-97967-0 }}

	~~{{PlanetMath attribution\|title=Montel's theorem\|id=5754}}~~

	~~[[Category:Compactness theorems]]~~
	~~[[Category:Theorems in complex analysis]]~~

96.5.29.8: /* Uniform electric field */

2013-09-04T18:50:31Z

Uniform electric field

Show changes

141.211.250.73: /* Uniform electric feild */

2012-08-28T13:45:15Z

Uniform electric feild

Show changes

Work (electrical) - Revision history

en>Wtshymanski: pompous "naturally"; lower case, ce; lower case in headings

en>Faizhaider: Disambiguated: Kirchhoff's Law → Kirchhoff's circuit laws

96.5.29.8: /* Uniform electric field */

141.211.250.73: /* Uniform electric feild */