Let \(f\) be holomorphic on a 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 \(g\colon D_\rho(z_0)\to\CC\) with \(f=g\) on \(D_\rho(z_0)\setminus\{z_0\}\).
Let \(f\) be holomorphic on \(D_r(z_0)\setminus\{z_0\}\). The following are equivalent.
\(z_0\) is a removable singularity of \(f\).
\(\lim_{z\to z_0}f(z)\) exists in \(\CC\).
\(f\) is bounded on \(D_\rho(z_0)\setminus\{z_0\}\) for some \(0<\rho\le r\).
There are an integer \(n\le0\) and a function \(h\) holomorphic on a neighborhood of \(z_0\) with \(f(z)=(z-z_0)^{-n}h(z)\) for \(z\neq z_0\) near \(z_0\).
(a)\(\Rightarrow\)(b): \(f(z)=g(z)\to g(z_0)\). (b)\(\Rightarrow\)(c): a function with a finite limit at \(z_0\) is bounded near \(z_0\). (c)\(\Rightarrow\)(a) is Riemann's removable singularity theorem. (a)\(\Rightarrow\)(d): take \(n=0\) and \(h=g\). (d)\(\Rightarrow\)(a): \(g(z)\coloneqq(z-z_0)^{-n}h(z)\) is holomorphic near \(z_0\) because \(-n\ge0\).
Condition (d) is the factorization \(f=(z-z_0)^{-n}h\) that defines a pole of order $n$ for \(n\ge1\), allowed here with \(n\le0\); in this sense a removable singularity is sometimes called a pole of order zero.
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FD-CCVUQRemovable singularity as a pole of order zerovariant-of