TITLE: Area integral near essential singularity
QUESTION [7 upvotes]: I'm studying for an exam and am stuck on the following.
If $f$ is holomorphic on the punctured unit disk $D- \{0\}$, and $0$ is an essential singularity does it follow that
$\displaystyle\int_{D -\{0\}} |f(z)|^{2} dA = \infty$

REPLY [5 votes]: I believe I found a solution to my own question.  I'll write it out to see if anyone agrees,   Basically we can write the integral as
$\displaystyle\int_{D -\{0\}} |f(z)|^{2} dA = 2\pi \int_{0}^{1} \sum_{n=-\infty}^{\infty} |a_{n}|^{2}r^{2n+1} dr$
where $a_{n}$ are the Laurent coeficients  Here I used Tonelli Thereom along with the fact that $\{e^{i \theta n} | n\in \mathbb{Z} \}$ are orthogonal.   
Since we have an essential singularity $a_{k} \ne 0$ for some $k<0$.  Thus we have 
$\displaystyle\int_{D -\{0\}} |f(z)|^{2} dA = \infty$