Removable singularity

An isolated singularity is removable when the function extends holomorphically across it, equivalently when its Laurent series has no negative-power terms.

A holomorphic function bounded on a punctured neighbourhood extends holomorphically across the puncture.

A holomorphic function tending to a finite limit at infinity becomes holomorphic at zero after reciprocal substitution.

If $f$ is holomorphic on $0<|z|<R$ andthen the singularity at zero is removable. The Laurent coefficient formula and Cauchy--Schwarz give $|a_{-k}|=O(r^k)$ for each $k\ge 1$, so every principal-part coefficient vanishes as $r\downarrow 0$.