PlanetMath: isolated singularity

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isolated singularity

(Definition)

Let $\mathbb{C}\cup\{\infty\}$ denote the Riemann sphere, and let $U\subset \mathbb{C}$ be open. Let $f\colon U \to \mathbb{C}\cup\{\infty\}$ be a function. We say that is an isolated singularity of if there exists an open set $V\subset U$ containing and such that is analytic on $V\!\smallsetminus\!\{z\}$ .

In other words, if we take the set of points in where is not analytic, the isolated singularities are exactly the isolated points of in the usual topological sense.

There are three kinds of isolated singularities:

removable singularities $\displaystyle \left( \text{e.g.\,} z = 0 \text{\, for the function\,} \frac{\sin{z}}{z} \right)$
poles $\displaystyle \left( \text{e.g.\,} z = 0 \text{\, for the function\,} \frac{1}{z^2} \right)$
essential singularities $\displaystyle \left( \text{e.g.\,} z = 0 \text{\, for the function\,} \exp{\frac{1}{z}} \right)$

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"isolated singularity" is owned by bwebste. [ full author list (5) | owner history (1) ]

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Cross-references: essential singularities, poles, removable singularities, isolated points, points, analytic, open set, function, open, Riemann sphere
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This is version 6 of isolated singularity, born on 2003-10-15, modified 2007-04-15.
Object id is 4939, canonical name is IsolatedSingularity.
Accessed 2194 times total.

Classification:

AMS MSC:

30-00 (Functions of a complex variable :: General reference works )

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