Ngô Quốc Anh

May 19, 2009

Casorati–Weierstrass theorem, the behavior of meromorphic functions near essential singularities

Filed under: Giải tích 7 (MA4247) — Tags: , — Ngô Quốc Anh @ 14:42

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

Formal statement of the theorem.

Start with some open subset U in the complex plane containing the number z_0, and a function f that is holomorphic on U\backslash \left\{ {{z_0}} \right\}, but has an essential singularity at z_0. The Casorati–Weierstrass theorem then states that

if V is any neighborhood of z_0 contained in U, then f(V \backslash \{z_0\}) is dense in \mathbb C.

This can also be stated as follows

for any \varepsilon > 0 and any complex number w, there exists a complex number z in U with |z-z_0| < \varepsilon and |f(z)-w| < \varepsilon.

Or in still more descriptive terms

f comes arbitrarily close to any complex value in every neighbourhood of z_0.

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) = e^\frac{1}{z} has an essential singularity at z_0 = 0, but the function g(z) = \frac{1}{z^3} does not (it has a pole at 0). Consider the function

\displaystyle f(z)=e^{1/z}.

This function has the following Laurent series about the essential singular point at z_0

\displaystyle f(z)=\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!z^{n}}.

(more…)

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