Casorati–Weierstrass theorem

Template:Short description In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati.<ref name="analysis">Template:Cite book</ref> In Russian literature it is called Sokhotski's theorem, because it was discovered independently by Sokhotski in 1868.<ref name="analysis"/>

Formal statement of the theorem

Start with some open subset <math>U</math> in the complex plane containing the number <math>z_0</math>, and a function <math>f</math> that is holomorphic on <math>U \setminus \{z_0\}</math>, but has an essential singularity at <math>z_0</math> . The Casorati–Weierstrass theorem then states that Template:Block indent

This can also be stated as follows: Template:Block indent

Or in still more descriptive terms: Template:Block indent

The theorem is considerably strengthened by Picard's great theorem, which states, in the notation above, that <math>f</math> assumes every complex value, with one possible exception, infinitely often on <math>V</math>.

In the case that <math>f</math> is an entire function and <math>a = \infty</math>, the theorem says that the values <math>f(z)</math> approach every complex number and <math>\infty</math>, as <math>z</math> tends to infinity. It is remarkable that this does not hold for holomorphic maps in higher dimensions, as the famous example of Pierre Fatou shows.<ref>Template:Cite journal , Template:Cite journal</ref>

File:Essential singularity.png

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).

Examples

The function Template:Math has an essential singularity at 0, but the function Template:Math does not (it has a pole at 0).

Consider the function <math display="block">f(z) = e^{1/z}.</math>

This function has the following Laurent series about the essential singular point at 0: <math display="block">f(z) = \sum_{n=0}^{\infty}\frac{1}{n!}z^{-n}.</math>

Because <math>f'(z) = - \frac{e^{{1}/{z}}}{z^2}</math> exists for all points Template:Math we know that Template:Math is analytic in a punctured neighborhood of Template:Math. Hence it is an isolated singularity, as well as being an essential singularity.

Using a change of variable to polar coordinates <math>z=re^{i \theta }</math> our function, Template:Math becomes: <math display="block">f(z)=e^{\frac{1}{r}e^{-i\theta}}=e^{\frac{1}{r}\cos(\theta)}e^{-\frac{1}{r}i \sin(\theta)}.</math>

Taking the absolute value of both sides: <math display="block">\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>

Thus, for values of θ such that Template:Math, we have <math>f(z) \to \infty</math> as <math>r \to 0</math>, and for <math>\cos \theta < 0</math>, <math>f(z) \to 0</math> as <math>r \to 0</math>.

Consider what happens, for example when z takes values on a circle of diameter Template:Math tangent to the imaginary axis. This circle is given by Template:Math. Then, <math display="block">f(z) = e^{R} \left[ \cos \left( R\tan \theta \right) - i \sin \left( R\tan \theta \right) \right] </math> and <math display="block">\left| f(z) \right| = e^R.</math>

Thus,<math>\left| f(z) \right|</math> may take any positive value other than zero by the appropriate choice of R. As <math>z \to 0</math> on the circle, <math display="inline"> \theta \to \frac{\pi}{2}</math> with R fixed. So this part of the equation: <math display="block">\left[ \cos \left( R \tan \theta \right) - i \sin \left( R \tan \theta \right) \right] </math> takes on all values on the unit circle infinitely often. Hence Template:Math takes on the value of every number in the complex plane except for zero infinitely often.

Proof of the theorem

A short proof of the theorem is as follows:

Take as given that function Template:Math is meromorphic on some punctured neighborhood Template:Math, and that Template:Math is an essential singularity. Assume by way of contradiction that some value Template:Mvar exists that the function can never get close to; that is: assume that there is some complex value Template:Mvar and some Template:Math such that Template:Math for all Template:Mvar in Template:Mvar at which Template:Mvar is defined.

Then the new function: <math display="block">g(z) = \frac{1}{f(z) - b}</math> must be holomorphic on Template:Math, with zeroes at the 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 Riemann's analytic continuation theorem. So the original function can be expressed in terms of Template:Math: <math display="block">f(z) = \frac{1}{g(z)} + b</math> for all arguments z in V \ {z₀}. Consider the two possible cases for <math display="block">\lim_{z \to z_0} g(z).</math>

If the limit is 0, then f has a pole at z₀ . If the limit is not 0, then z₀ is a removable singularity of f . Both possibilities contradict the assumption that the point z₀ is an essential singularity 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.<ref name="CV">Template:Cite book</ref> 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).<ref name="BB">Template:Cite book</ref> However, Briot and Bouquet removed this theorem from the second edition (1875).

References

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• Section 31, Theorem 2 (pp. 124–125) of Template:Citation