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. Formal statement of the theorem Start with some open subset in the complex plane containing the number , and a function that is holomorphic on , but has an essential singularity at . 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 assumes every complex value, with one possible exception, infinitely often on . In the case that is an entire function and , the theorem says that the values approach every complex number and , as 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. Examples The function has an essential singularity at 0, but the function does not (it has a pole at 0). Consider the function This function has the following Taylor series about the essential singular point at 0: Because 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. Using a change of variable to polar coordinates our function, becomes: Taking the absolute value of both sides: Thus, for values of θ such that , we have as , and for , as . 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, and Thus, may take any positive value other than zero by the appropriate choice of R. As on the circle, with R fixed. So this part of the equation: 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: