30-00-IsolatedSingularity.tex

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\pmtitle{isolated singularity}
\pmrecord{10}{34939}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{30-00}

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\begin{document}
%\section*{isolated singularity}
%\begin{123listcolonstyle} 
%\item $f:U \subset \C \rightarrow \C\cup\{\infty\}$ 
%\item $z_0\in U$ 
%\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/}.}

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\, $V\!\smallsetminus\!\{z\}$. 

In other \PMlinkescapetext{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.\\

There are three kinds of isolated singularities:
\begin{itemize}
\item removable singularities $\displaystyle \left( \text{e.g.\;\;} z = 0 \text{\; for the function\,} \frac{\sin{z}}{z} \right)$
\item poles $\displaystyle \left( \text{e.g.\;\;} z = 0 \text{\; for the function\,} \frac{1}{z^2} \right)$
\item essential singularities $\displaystyle \left( \text{e.g.\;\;} z = 0 \text{\; for the function\,} \exp{\frac{1}{z}} \right)$
\end{itemize}
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\end{document}