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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. | |
| 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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We all want consume healthier and lose weight, but whenever it gets down to it, the number of information that you could buy is completely overwhelming. There doesn't seem to be one "right" way and many expert reports are contradictory and puzzling. On top of this, even though most people know it doesn't work, they've got a particular body part they want to focus for. What they would love are diet tips for six pack abs. But what they find out is how the world of nutrition is filled with trendy fad diets, from low fat and lower carb to the soup diet and stomach fat diet.
Avoid getting hungry. To do this, you should go on a diet where you're eating smaller healthy meals from the throughout the day. Also, it's important that protein and fiber is that are part of every meal to help combat hunger pangs.
Take care that you have to avoid view meals journal as being a chore. Your entries end up being basic as well as simple to understand to a person will. You just want to know exactly what your eating habits are like to identify how you can make smarter choices and reduced fat deprivation. Ideally, it shouldn't take five minutes (max) each and every day to journal quickly as well as you eat something. After breakfast, have a few seconds to make note of what you ate. Exactly how much you had. And do that every time you snack, enjoy a meal both at home and in neighborhood.
The experts advise that folks can use yogurt in order to create masks (dry skin use nutribullet full-fat yogurt and oily skin use low-fat yogurt). The nutrition in the yogurt can make the skin more neat and bright.
You furthermore heard from various involving how it's okay to use "good carbs". Most people probably don't understand what a "carbohydrate" is set in the first place, not to mention a "good carb". Let's take a minute and explore the differences of 2 food sources as it pertains to weight dissapointment.
KB: Your product comes previously flavors of Pomegranate, Mangosteen, and Green Tea-Lemon. Far from the orange and lemon-lime staples of drink mix. Why the exotic palate, the actual exactly is really a mangosteen?
Finally, if you have the guts and wish to put yourself in optimum position accomplish them - make them public! Create a copy of the goals and share with it together with trusted friend or relative - someone you've labeled as part of one's support solution. There's nothing like a little accountability to keep you focused!