# Weierstrass–Casorati theorem

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In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of meromorphic functions near essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati.

## Contents

### Formal statement of the theorem

Start with some open subset U in the complex plane containing the number z0, and a function f that is holomorphic on U \ {z0}, but has an essential singularity at z0 . The Casorati–Weierstrass theorem then states that

This can also be stated as follows:

Or in still more descriptive terms:

This form of the theorem also applies if f is only meromorphic.

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.

### Examples

The function f(z) = exp(1/z) has an essential singularity at z0 = 0, but the function g(z) = 1/z3 does not (it has a pole at 0).

Consider the function

This function has the following Laurent series about the essential singular point at z0:

Because $f'(z) =\frac{-e^{\frac{1}{z}}}{z^{2}}$ exists for all points z ≠ 0 we know that ƒ(z) is analytic in the punctured neighborhood of z = 0. Hence it is an isolated singularity, as well as being an essential singularity.

Using a change of variable to polar coordinates z = reiθ our function, ƒ(z) = e1/z becomes:

Taking the absolute value of both sides:

Thus, for values of θ such that cos θ > 0, we have $f(z)\rightarrow\infty$ as $r \rightarrow 0$, and for cosθ < 0, $f(z) \rightarrow 0$ as $r \rightarrow 0$.

Consider what happens, for example when z takes values on a circle of diameter 1/R tangent to the imaginary axis. This circle is given by r = (1/R) cos θ. Then,