# Lectures and Notes

## week 8 notes review

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.

## Isolated Singularities

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.

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

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

3. 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 $$z_0$$.

$$0 < |z| < 2pi$$