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. In Russian literature it is called Sokhotski's theorem, because it was discovered independently by Sokhotski in 1868.
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).
Examples
The function has an essential singularity at 0, but the function 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^}{z^2}</math> 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. <!-- (a pole that is a cluster point of poles is essential, hence false remark:) like all other essential singularities. -->
Using a change of variable to polar coordinates <math>z=re^{i \theta }</math> our function, 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 , 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 tangent to the imaginary axis. This circle is given by . 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 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 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:
<math display="block">g(z) = \frac{1}{f(z) - b}</math>
must be holomorphic on , 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 :
<math display="block">f(z) = \frac{1}{g(z)} + b</math>
for all arguments z in V \ {z<sub>0</sub>}. 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 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.
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).