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removable singularity (Definition)

Let $ U\subset\mathbb{C}$ be an open neighbourhood of a point $ a\in \mathbb{C}$. We say that a function $ f:U\backslash\{a\}\rightarrow \mathbb{C}$ has a removable singularity at $ a$, if the complex derivative $ f'(z)$ exists for all $ z\neq a$, and if $ f(z)$ is bounded near $ a$.

Removable singularities can, as the name suggests, be removed.

Theorem 1   Suppose that $ f:U\backslash\{a\}\rightarrow \mathbb{C}$ has a removable singularity at $ a$. Then, $ f(z)$ can be holomorphically extended to all of $ U$, i.e. there exists a holomorphic $ g:U\rightarrow\mathbb{C}$ such that $ g(z)=f(z)$ for all $ z\neq a$.

Proof. Let $ C$ be a circle centered at $ a$, oriented counterclockwise, and sufficiently small so that $ C$ and its interior are contained in $ U$. For $ z$ in the interior of $ C$, set

$\displaystyle g(z) = \frac{1}{2\pi i} \oint_C \frac{f(\zeta)}{\zeta-z}d\zeta.$
Since $ C$ is a compact set, the defining limit for the derivative
$\displaystyle \frac{d}{dz} \frac{f(\zeta)}{\zeta-z}= \frac{f(\zeta)}{(\zeta-z)^2}$
converges uniformly for $ \zeta\in C$. Thanks to the uniform convergence, the order of the derivative and the integral operations can be interchanged. Hence, we may deduce that $ g'(z)$ exists for all $ z$ in the interior of $ C$. Furthermore, by the Cauchy integral formula we have that $ f(z)=g(z)$ for all $ z\neq a$, and therefore $ g(z)$ furnishes us with the desired extension.



"removable singularity" is owned by rmilson.
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See Also: essential singularity
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Cross-references: extension, Cauchy integral formula, operations, integral, order, uniform convergence, converges uniformly, derivative, limit, compact set, contained, interior, oriented, circle, proof, holomorphic, near, bounded, complex derivative, function, point, neighbourhood, open
There are 9 references to this entry.

This is version 2 of removable singularity, born on 2002-08-13, modified 2003-03-29.
Object id is 3289, canonical name is RemovableSingularity.
Accessed 4106 times total.

Classification:
AMS MSC: 30E99 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Miscellaneous)
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