Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 12F / b / Solution

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 $r\downarrow0$.

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 Incoming links: Uniform L2 circle bound for a removable singularity