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Taylors Theorem says if $f$ is analytic on $\{z:|z-z_0|  Cauchys inequality says that if \f is analytic in $\{z\in\mathbb{c}:|z-z_0|  \subsection{Isolated Singularities}  a function which is analytic on the punctured disk \{z\matbb{c}:0<|z-z_0| \{z\in\mathbb{c}:0<|z-z_0|  has an isolated singularity at $z_0$