Laurent Series¶

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 \(z∈D(a,r)\).