authorea.com/5280

Taylors Theorem says if \(f\) is analytic on \(\{z:|z-z_0|<r\}\) and continuous on the domain that includes the boundary, then \(f(z)=\sum_{n=0}^{\infty}f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}\) and this series converges absolutely. Cauchys inequality says that if is analytic in \(\{z\in\mathbb{c}:|z-z_0|<r\}\) and \(|f(z)|\leq c\) in the disk then the function converges absolutely.

A function which is analytic on the punctured disk \(\{z\in\mathbb{c}:0<|z-z_0|<r\}\) has an isolated singularity at \(z_0\).There are three examples of isolated singularities.

removable singularity: where \(f(z)\) is bounded for some \(r>0\) on \(\{0<|z-z_0|<r\}\), remains bounded as \(z\rightarrow z_0\)

poles: where \(\lim_{z\to z_0}|f(z)|=\infty\)

essential singularity: when 1 or 2 dont apply

lemma: if \(f\) has a removable singularity at \(z_0\) then the \(\lim_{z\to z_0} f(z)\) exists and extends \(f\) to an analytic function at \(z_0\).

\(0 < |z| < 2pi\)

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