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,

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