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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.

user13866
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2 Answers2

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The one with the $|f(z)|^2$ is simpler. Consider the Laurent series $f(z) = \sum_{n=-\infty}^\infty c_n z^n$. By the orthogonality of the functions $e^{in\theta}$ on $[0, 2\pi]$, we have $\int_0^{2\pi} |f(r e^{i\theta})|^2 \ d\theta = 2 \pi \sum_{n=-\infty}^\infty |c_n|^2 r^{2n}$ and $\int_D |f(z)|^2 \ dA = \int_0^1 \int_0^{2\pi} r |f(r e^{i\theta}|^2 \ d\theta \ dr$. Since $\int_0^1 r^{2n+1}\ dr$ is finite for $n \ge 0$ and infinite for $n \le -1$, the only way to have $\int_D |f(z)|^2 \ dA < \infty$ is that all the coefficients for $n \le -1$ are 0.

Robert Israel
  • 448,999
  • In the middle of the process, we kind of need to interchange the infinite series with integral, what domain did you choose to ensure the Laurent series to be convergent absolutely? – JacobsonRadical Dec 06 '20 at 01:19
  • For any function that is analytic on an open annulus $a < |z| < b$, the Laurent series converges absolutely on that annulus. – Robert Israel Dec 06 '20 at 03:11
  • so for example here we actually consider $0<|z|<1$. What i know is that then we have the Laurent series converges absolutely on $r_{1}\leq |z|\leq r_{2}$ for any $r_{1}>0$ and $r_{2}<1$. So we can restrict the annulus as $\epsilon\leq |z|\leq 1-\epsilon$, which makes our Laurent series converges absolutely and uniformly. But then the outside integral should be then $\lim_{\epsilon\rightarrow 0}\int_{\epsilon}^{1-\epsilon}$, I am not sure how to interchange $$\lim_{\epsilon\rightarrow 0}\int_{\epsilon}^{1-\epsilon}\sum_{n=1}^{\infty}|c_{n}|^{2}r^{2n+1}dr$$ – JacobsonRadical Dec 06 '20 at 03:29
  • Monotone convergence theorem. – Robert Israel Dec 06 '20 at 06:43
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For the first problem, write $c_n$ as an integral around the circle of radius $r$ centred at $0$. If $\int_D |f(z)|\ dA < \infty$, the integral of that times a certain power of $r$ for $r$ from 0 to 1 would be finite ...

Robert Israel
  • 448,999