Casorati–Weierstrass Theorem
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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem, because it was discovered independently by Sokhotski in 1868.


Formal statement of the theorem

Start with some
open subset In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...
U in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
containing the number z_0, and a function f that is
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...
on U \setminus \, but has an
essential singularity In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities t ...
at z_0 . The ''Casorati–Weierstrass theorem'' then states that This can also be stated as follows: Or in still more descriptive terms: 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. In the case that f is an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...
and a = \infty, the theorem says that the values f(z) approach every complex number and \infty, as z tends to infinity. It is remarkable that this does not hold for
holomorphic map In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex der ...
s in higher dimensions, as the famous example of
Pierre Fatou Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of mathematical analysis, analysis. The Fatou lemma and the Fatou set are named aft ...
shows. ,


Examples

The function has an essential singularity at 0, but the function does not (it has a pole at 0). Consider the function f(z) = e^. This function has the following
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
about the essential singular point at 0: f(z) = \sum_^\fracz^. Because f'(z) = - \frac exists for all points we know that is analytic in a punctured neighborhood of . Hence it is an isolated singularity, as well as being an
essential singularity In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities t ...
. Using a change of variable to
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
z=re^ our function, becomes: f(z)=e^=e^e^. Taking the
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of both sides: \left, f(z) \ = \left, e^ \ \left, e^ \right , =e^. Thus, for values of ''θ'' such that , we have f(z) \to \infty as r \to 0, and for \cos \theta < 0, f(z) \to 0 as r \to 0. Consider what happens, for example when ''z'' takes values on a circle of diameter tangent to the imaginary axis. This circle is given by . Then, f(z) = e^ \left \cos \left( R\tan \theta \right) - i \sin \left( R\tan \theta \right) \right and \left, f(z) \ = e^R. Thus,\left, f(z) \ may take any positive value other than zero by the appropriate choice of ''R''. As z \to 0 on the circle, \theta \to \frac with ''R'' fixed. So this part of the equation: \left \cos \left( R \tan \theta \right) - i \sin \left( R \tan \theta \right) \right takes on all values on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
infinitely often. Hence takes on the value of every number in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
except for zero infinitely often.


Proof of the theorem

A short proof of the theorem is as follows: Take as given that function is meromorphic on some punctured neighborhood , and that is an essential singularity. Assume by way of contradiction that some value exists that the function can never get close to; that is: assume that there is some complex value and some such that for all in at which is defined. Then the new function: g(z) = \frac must be holomorphic on , with zeroes at the
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of ''f'', and bounded by 1/ε. It can therefore be analytically continued (or continuously extended, or holomorphically extended) to ''all'' of ''V'' by Riemann's analytic continuation theorem. So the original function can be expressed in terms of : f(z) = \frac + b for all arguments ''z'' in ''V'' \ . Consider the two possible cases for \lim_ g(z). If the limit is 0, then ''f'' has a pole at ''z''0 . If the limit is not 0, then ''z''0 is a
removable singularity In complex analysis, a removable singularity of a holomorphic function is a point at which the function is Undefined (mathematics), undefined, but it is possible to redefine the function at that point in such a way that the resulting function is ...
of ''f'' . Both possibilities contradict the assumption that the point ''z''0 is an
essential singularity In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities t ...
of the function ''f'' . Hence the assumption is false and the theorem holds.


History

The history of this important theorem is described by Collingwood and Lohwater. It was published by Weierstrass in 1876 (in German) and by Sokhotski in 1868 in his Master thesis (in Russian). So it was called Sokhotski's theorem in the Russian literature and Weierstrass's theorem in the Western literature. The same theorem was published by Casorati in 1868, and by Briot and Bouquet in the ''first edition'' of their book (1859). However, Briot and Bouquet ''removed'' this theorem from the second edition (1875).


References

* Section 31, Theorem 2 (pp. 124–125) of {{DEFAULTSORT:Casorati-Weierstrass theorem Theorems in complex analysis Articles containing proofs