Let \(z_0\in\CC\), let \(r>0\), and let \(f\) be holomorphic on the punctured disc \(D_r(z_0)\setminus\{z_0\}\). The point \(z_0\) is a removable singularity of \(f\) if there exist \(0<\rho\le r\) and a holomorphic function \(g\colon D_\rho(z_0)\to\CC\) such that \(f(z)=g(z)\) for \(0<\abs{z-z_0}<\rho\).
(Riemann's removable singularity theorem)
Let \(f\) be holomorphic on \(D_r(z_0)\setminus\{z_0\}\). The following are equivalent.
\(z_0\) is a removable singularity of \(f\).
\(\lim_{z\to z_0}(z-z_0)f(z)=0\).
\(f\) is bounded on \(D_\rho(z_0)\setminus\{z_0\}\) for some \(0<\rho\le r\).
(a)\(\Rightarrow\)(c). The function \(g\) is continuous on \(D_\rho(z_0)\), hence bounded on \(D_{\rho/2}(z_0)\), and \(f=g\) there away from \(z_0\).
(c)\(\Rightarrow\)(b). If \(\abs{f}\le M\) near \(z_0\), then \(\abs{(z-z_0)f(z)}\le M\abs{z-z_0}\to0\).
(b)\(\Rightarrow\)(a). Define \(h(z)\coloneqq(z-z_0)^2f(z)\) for \(0<\abs{z-z_0}<r\) and \(h(z_0)\coloneqq0\). Then \(h\) is holomorphic away from \(z_0\), and at \(z_0\) \[ \frac{h(z)-h(z_0)}{z-z_0}=(z-z_0)f(z)\to0, \] so \(h\) is holomorphic on \(D_r(z_0)\) with \(h(z_0)=h'(z_0)=0\). Its Taylor expansion is therefore \(h(z)=\sum_{k\ge2}a_k(z-z_0)^k\) on \(D_r(z_0)\), and \(g(z)\coloneqq\sum_{k\ge2}a_k(z-z_0)^{k-2}\) is holomorphic on \(D_r(z_0)\) and equals \(f\) on the punctured disc.
In terms of the Laurent expansion \(f(z)=\sum_{k\in\ZZ}c_k(z-z_0)^k\) on \(D_r(z_0)\setminus\{z_0\}\), the point \(z_0\) is a removable singularity if and only if \(c_k=0\) for all \(k<0\), that is, if and only if the order \(v_{z_0}(f)\) satisfies \(v_{z_0}(f)\ge0\).