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[parent] proof of Casorati-Weierstrass theorem (Proof)

Assume that $ a$ is an essential singularity of $ f$. Let $ V\subset U$ be a punctured neighborhood of $ a$, and let $ \lambda\in\mathbb{C}$. We have to show that $ \lambda$ is a limit point of $ f(V)$. Suppose it is not, then there is an $ \epsilon>0$ such that $ \vert f(z)-\lambda\vert>\epsilon$ for all $ z\in V$, and the function

$\displaystyle g:V\to\mathbb{C}, z\mapsto\frac{1}{f(z)-\lambda} $
is bounded, since $ \vert g(z)\vert=\frac{1}{\vert f(z)-\lambda\vert}<\epsilon^{-1}$ for all $ z\in V$. According to Riemann's removable singularity theorem, this implies that $ a$ is a removable singularity of $ g$, so that $ g$ can be extended to a holomorphic function $ \bar g:V\cup\{a\}\to\mathbb{C}$. Now
$\displaystyle f(z)=\frac{1}{\bar g(z)}-\lambda $
for $ z\neq a$, and $ a$ is either a removable singularity of $ f$ (if $ \bar g(z)\neq 0$) or a pole of order $ n$ (if $ \bar g$ has a zero of order $ n$ at $ a$). This contradicts our assumption that $ a$ is an essential singularity, which means that $ \lambda$ must be a limit point of $ f(V)$. The argument holds for all $ \lambda\in\mathbb{C}$, so $ f(V)$ is dense in $ \mathbb{C}$ for any punctured neighborhood $ V$ of $ a$.

To prove the converse, assume that $ f(V)$ is dense in $ \mathbb{C}$ for any punctured neighborhood $ V$ of $ a$. If $ a$ is a removable singularity, then $ f$ is bounded near $ a$, and if $ a$ is a pole, $ f(z)\to\infty$ as $ z\to a$. Either of these possibilities contradicts the assumption that the image of any punctured neighborhood of $ a$ under $ f$ is dense in $ \mathbb{C}$, so $ a$ must be an essential singularity of $ f$.



"proof of Casorati-Weierstrass theorem" is owned by pbruin.
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See Also: Picard's theorem


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Cross-references: image, near, converse, dense in, argument, zero of order, order, pole, holomorphic function, removable singularity, implies, Riemann's removable singularity theorem, bounded, function, limit point, neighborhood, essential singularity

This is version 1 of proof of Casorati-Weierstrass theorem, born on 2003-04-03.
Object id is 4144, canonical name is ProofOfCasoratiWeierstrassTheorem.
Accessed 3246 times total.

Classification:
AMS MSC: 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)
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