Laurent Series

This thorem says that any analytic function in an annulus region has series expansion consist of positive and negative powers of $z$.

Taylor’s Thorem: Let $\Omega$ be an annulus formed by two concentric circle of radius $r_1$ and $r_2$ with center $a$ and $r_1 < r_2$. open connected set and let $a\in \Omega$. Let $f:\Omega→\mathbb{C}$ be analytic on $\Omega$. Then there exists an $r>0$ such that \(f(z)=\sum_{n=0}^\infty \dfrac{f^{(n)}(a)}{n!}(z−a)^n; \quad z ∈ D(a,r).\) In other words, the Taylor series of $f$ at $a$ converges uniformly and absolutely to $f$ for all $z∈D(a,r)$.