# 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 \(z∈D(a,r)\).

Last modified on: 2023-01-04 23:20:05