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| In [[complex analysis]], a branch of mathematics, the '''Casorati–Weierstrass theorem''' describes the behaviour of [[holomorphic function]]s near their [[essential singularity|essential singularities]]. It is named for [[Karl Theodor Wilhelm Weierstrass]] and [[Felice Casorati (mathematician)|Felice Casorati]]. In Russian literature it is called [[Yulian Sokhotski|Sokhotski's]] theorem.
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| ==Formal statement of the theorem==
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| Start with some [[open set|open subset]] ''U'' in the [[complex number|complex plane]] containing the number <math>z_0</math>, and a function ''f'' that is [[holomorphic function|holomorphic]] on <math>U\ \backslash\ \{z_0\}</math>, but has an [[essential singularity]] at <math>z_0</math> . The ''Casorati–Weierstrass theorem'' then states that
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| :if ''V'' is any [[Neighbourhood (mathematics)|neighbourhood]] of <math>z_0</math> contained in ''U'', then <math>f(V\ \backslash\ \{z_0\})</math> is [[dense set|dense]] in '''C'''.
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| This can also be stated as follows:
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| :for any ε > 0, δ >0, and complex number ''w'', there exists a complex number ''z'' in ''U'' with |''z'' − <math>z_0</math>| < δ and |''f''(''z'') − ''w''| < ε .
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| Or in still more descriptive terms:
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| :''f'' comes arbitrarily close to ''any'' complex value in every neighbourhood of <math>z_0</math>.
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| This form of the theorem also applies if ''f'' is only [[meromorphic]].
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| The theorem is considerably strengthened by [[Picard's great theorem]], which states, in the notation above, that ''f'' assumes ''every'' complex value, with one possible exception, infinitely often on ''V''.
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| In the case that ''f'' is an [[entire function]] and ''a=∞'', the theorem says that the values ''f(z)''
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| approach every complex number and ''∞'', as ''z'' tends to infinity.
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| It is remarkable that this does not hold for [[holomorphic map]]s in higher dimensions,
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| as the famous example of [[Pierre Fatou]] shows.<ref>{{cite article|first=P.|last=Fatou|title=Sur les fonctions
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| meromorphes de deux variables|journal=Comptes rendus|volume=175|year=1922|pages=862,1030.}}</ref>
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| [[Image:Essential singularity.png|right|220px|thumb|Plot of the function exp(1/''z''), centered on the essential singularity at ''z'' = 0. The hue represents the complex argument, the luminance represents the absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which would be uniformly white).]]
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| ==Examples==
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| The function ''f''(''z'') = [[exponential function|exp]](1/''z'') has an essential singularity at 0, but the function ''g''(''z'') = 1/''z''<sup>3</sup> does not (it has a [[pole (complex analysis)|pole]] at 0).
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| Consider the function
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| : <math>f(z)=e^{1/z}.\,</math>
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| This function has the following [[Laurent series]] about the [[essential singularity|essential singular point]] at 0:
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| : <math>f(z)=\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}z^{-n}.</math>
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| Because <math>f'(z) =\frac{-e^{\frac{1}{z}}}{z^{2}}</math> exists for all points ''z'' ≠ 0 we know that ''ƒ''(''z'') is analytic in a [[punctured neighborhood]] of ''z'' = 0. Hence it is an [[isolated singularity]], as well as being an [[essential singularity]]. <!-- (a pole that is a cluster point of poles is essential, hence false remark:) like all other essential singularities. -->
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| Using a change of variable to [[polar coordinates]] <math>z=re^{i \theta }</math> our function, ''ƒ''(''z'') = ''e''<sup>1/''z''</sup> becomes:
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| : <math>f(z)=e^{\frac{1}{r}e^{-i\theta}}=e^{\frac{1}{r}\cos(\theta)}e^{-\frac{1}{r}i \sin(\theta)}.</math>
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| Taking the [[absolute value]] of both sides:
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| : <math>\left| f(z) \right| = \left| e^{\frac{1}{r}\cos \theta} \right| \left| e^{-\frac{1}{r}i \sin(\theta)} \right | =e^{\frac{1}{r}\cos \theta}.</math>
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| Thus, for values of ''θ'' such that cos ''θ'' > 0, we have <math>f(z)\rightarrow\infty</math> as <math>r \rightarrow 0</math>, and for <math>\cos \theta <0</math>, <math>f(z) \rightarrow 0</math> as <math>r \rightarrow 0</math>.
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| Consider what happens, for example when ''z'' takes values on a circle of diameter 1/''R'' tangent to the imaginary axis. This circle is given by ''r'' = (1/''R'') cos ''θ''. Then,
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| : <math>f(z) = e^{R} \left[ \cos \left( R\tan \theta \right) - i \sin \left( R\tan \theta \right) \right] </math>
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| and
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| : <math>\left| f(z) \right| = e^R.\,</math>
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| Thus,<math>\left| f(z) \right|</math> may take any positive value other than zero by the appropriate choice of ''R''. As <math>z \rightarrow 0</math> on the circle, <math> \theta \rightarrow \frac{\pi}{2}</math> with ''R'' fixed. So this part of the equation:
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| : <math>\left[ \cos \left( R \tan \theta \right) - i \sin \left( R \tan \theta \right) \right] \, </math>
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| takes on all values on the [[unit circle]] infinitely often. Hence ''f''(''z'') takes on the value of every number in the [[complex plane]] except for zero infinitely often.
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| ==Proof of the theorem==
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| A short proof of the theorem is as follows:
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| Take as given that function ''f'' is [[meromorphic function|meromorphic]] on some punctured neighborhood ''V'' \ {''z''<sub>0</sub>}, and that ''z''<sub>0</sub> is an essential singularity. Assume by way of contradiction that some value ''b'' exists that the function can never get close to; that is: assume that there is some complex value ''b'' and some ε > 0 such that |''f''(''z'') − ''b''| ≥ ε for all ''z'' in ''V'' at which ''f'' is defined.
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| Then the new function:
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| :<math>g(z) = \frac{1}{f(z) - b}</math> | |
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| must be holomorphic on ''V'' \ {''z''<sub>0</sub>}, with [[Zero (complex analysis)|zeroes]] at the [[Pole (complex analysis)|poles]] of ''f'', and bounded by 1/ε. It can therefore be analytically continued (or continuously extended, or holomorphically extended) to ''all'' of ''V'' by [[Removable singularity#Riemann's theorem|Riemann's analytic continuation theorem]]. So the original function can be expressed in terms of ''g'':
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| :<math>f(z) = \frac{1}{g(z)} + b</math>
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| for all arguments ''z'' in ''V'' \ {''z''<sub>0</sub>}. Consider the two possible cases for
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| :<math>\lim_{z \to z_0} g(z).</math>
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| If the limit is 0, then ''f'' has a [[Pole (complex analysis)|pole]] at ''z''<sub>0</sub> . If the limit is not 0, then ''z''<sub>0</sub> is a [[removable singularity]] of ''f'' . Both possibilities contradict the assumption that the point ''z''<sub>0</sub> is an [[essential singularity]] of the function ''f'' . Hence the assumption is false and the theorem holds.
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| ==History==
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| The history of this important theorem is described by
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| [[Edward Collingwood|Collingwood]] and Lohwater.<ref name="CV">{{cite book|first1=E|last1=Collingwood|first2=A |last2=Lohwater|title=The theory of cluster sets|
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| publisher=[[Cambridge University Press]]|year=1966}}</ref>
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| It was published by Weierstrass in 1876 (in German) and by Sokhotski in 1873 (in Russian).
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| So it was called Sokhotski's theorem in the Russian literature and Weierstrass's theorem in
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| the Western literature.
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| The same theorem was published by Casorati in 1868, and
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| by Briot and Bouquet in the ''first edition'' of their book (1859).<ref name="BB">{{cite book|first1=Ch|last1= Briot|
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| first2=C|last2=Bouquet|
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| title=Theorie des fonctions doublement periodiques, et en particulier, des fonctions elliptiques|place=Paris|year=1859}}</ref>
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| However, Briot and Bouquet ''removed'' this theorem from the second edition (1875).
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| ==References==
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| <references />
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| * Section 31, Theorem 2 (pp. 124–125) of {{Citation
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| | last=Knopp
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| | first=Konrad
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| | author-link=Konrad Knopp
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| | title=Theory of Functions
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| | publisher=[[Dover Publications]]
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| | year=1996
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| | isbn=978-0-486-69219-7
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| }}
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Essential_singular_point Essential singularity] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Casorati-Sokhotskii-Weierstrass_theorem Casorati-Weierstrass theorem] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| {{DEFAULTSORT:Casorati-Weierstrass theorem}}
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| [[Category:Complex analysis]]
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| [[Category:Theorems in complex analysis]]
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| [[Category:Articles containing proofs]]
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The one wrote post is called Tawanda Tally and she believes appears a little bit quite quite. Ohio is where me and my wife live. One of the finest things in the world for him is playing mah jongg but he is struggling you are able to time so as. Booking holidays is the place she supports her folks. My husband and I maintain website. You might want to confirm it out here: http://webbook.tk/blogs/viewstory/9188
my web site; nutribullet