Uniform L2 circle bound for a removable singularity
= Uniform L2 circle bound for a removable singularity
If $f$ is holomorphic on $0<|z|<R$ and
$$
\sup_{0<r<R}\int_0^{2\pi}|f(re^{i\theta})|^2\,d\theta<\infty,
$$
then the singularity at zero is removable. The Laurent coefficient formula and Cauchy--Schwarz give $|a_{-k}|=O(r^k)$ for each $k\geq1$, so every principal-part coefficient vanishes as $r\downarrow0$.