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| %\section*{isolated singularity} |
%\section*{isolated singularity} |
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| %\item $f:U \subset \C \rightarrow \C\cup\{\infty\}$ |
%\item $f:U \subset \C \rightarrow \C\cup\{\infty\}$ |
| %\item $z_0\in U$ |
%\item $z_0\in U$ |
| %\item $f$ analytic on $U \setminus \{z_0\}$ |
%\item $f$ analytic on $U \setminus \{z_0\}$ |
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%{\it Note: This is a ``seed'' entry written using a short-hand format described in \htmladdnormallink{this FAQ}{http://www.ma.utexas.edu/~jcorneli/h/FAQ/}.} |
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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 $z$ is an \emph{isolated singularity} of $f$ if there exists an open set $V\subset U$ containing $z$ and such that $f$ is analytic on\, $V\!\smallsetminus\!\{z\}$.
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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 $z$ is an \emph{isolated singularity} of $f$ if there exists an open set $V\subset U$ containing $z$ and such that $f$ is analytic on $V\setminus\{z\}$.
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| In other words, if we take the set $S$ of points in $U$ where $f$ is \emph{not} analytic, the isolated singularities are exactly the isolated points of $S$ in the usual topological sense. |
In other words, if we take the set $S$ of points in $U$ where $f$ is \emph{not} analytic, the isolated singularities are exactly the isolated points of $S$ in the usual topological sense. |
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| There are three kinds of isolated singularities: |
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| \begin{itemize} |
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| \item removable singularities (e.g.\, $z = 0$\, for the function \, $\frac{\sin{z}}{z}$) |
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| \item poles (e.g.\, $z = 0$\, for the function\,$\frac{1}{z^2}$) |
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| \item essential singularities (e.g.\, $z = 0$\, forthe function\, $\exp{\frac{1}{z}}$) |
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| \end{itemize} |
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