Casorati–Weierstrass theorem
   HOME

TheInfoList



OR:

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, 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 derivativ ...
s near their
essential singularities In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
. It is named for Karl Theodor Wilhelm Weierstrass and
Felice Casorati Felice Casorati (December 4, 1883 – March 1, 1963) was an Italian painter, sculptor, and printmaker. The paintings for which he is most noted include figure compositions, portraits and still lifes, which are often distinguished by unusua ...
. In Russian literature it is called Sokhotski's theorem.


Formal statement of the theorem

Start with some
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
U in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
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 derivativ ...
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 odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
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 In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb is ...
, 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 fin ...
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 derivativ ...
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 analysis. The Fatou lemma and the Fatou set are named after him. Biography P ...
shows. ,


Examples

The function has an essential singularity at 0, but the function does not (it has a
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
at 0). Consider the function f(z) = e^. This function has the following
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
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 In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
of . Hence it is an
isolated singularity In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number ''z0'' is an isolated singularity of a function ''f'' if there exists an open disk ''D'' ...
, 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 odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
. Using a change of variable to
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
z=re^ our function, becomes: f(z)=e^=e^e^. Taking the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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 Eucl ...
infinitely often. Hence takes on the value of every number in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
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 In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), poles ...
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 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
at the
poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
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 Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
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, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourh ...
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 odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
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