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I'm having trouble with the following problem:

Let $f$ be holomorphic on the punctured unit disc, $D$. If $\int_D|f(z)|dA(z)<\infty$, then $z=0$ is either a removable singularity or a simple pole of $f$.

A similar problem is to prove or disprove that if $\int_D|f(z)|^2dA(z)<\infty$, then $f$ has a removable singularity at $0$.

I've tried using the Mean Value Property for $f$ and taking the limit as $z\to0$, but I didn't get anywhere.

  • 1
    Have you tried integrating $1/|z|^2$ over the disk?2011-09-11
  • 0
    Matt E: isn't that infinity?2011-09-11

2 Answers 2