Looser Conditions on Casorati-Weierstrass The Casorati-Weierstrass Theorem presented in Stein and Shakarchi's "Complex Analysis" discusses the behavior of the image of a homlomorphic function in a punctured disc about an essential singularity. 
I show that the function need not be holomorphic (or meromorphic) as long as its infinitely many poles converge to the essential singularity. In this way, I think we can put a "looser" condition of Casorati-Weierstrass. 
I know this is bad form to just post links to read stuff, but I prove it here: http://www.princeton.edu/~rghanta/Casorati-Weierstrass_2.pdf It is a two page write-up, and look on the second page for my proof! 
I don't think this result is that big of a deal, but I am wondering if anyone has seen this result before cited in another book? If so, where can I look. 
And now to the primary reason for posting this: 
Since I have already descended down this route and I find essential singularities very interesting, do you have any suggestions of where I can go from what I have shown? Do you recommend any texts or papers that I can read to better understand behavior near an essential singularity. I am already aware of Picard's Theorem, but I'm also interested if there is anything else we can say about essential singularities. 
I appreciate your help!
 A: It seems to me that the proof of C-W carries over to this new situation. Namely, let $(x_j)$ for $j \geq 1$ be the poles of $f$, and let $z_0$ be the essential singularity.
Suppose the image of $f$ is not dense in $\mathbb C$, and pick a point $w$ such that $|f(z) - w| > \delta$ for all $z$ in $D \setminus \{z_0\}$, while waving hands to take care of the poles.
Now define a function $g$ on $D \setminus \{z_0, x_1, x_2, \ldots\}$ by $g(z) = 1/(f(z) - w)$. It is bounded in a neighborhood of each pole $x_j$, so it extends to a holomorphic function on $D \setminus \{z_0\}$. Like in C-W, the extension is bounded by $\delta$ on $D$, which contradicts that $z_0$ is an essential singularity.
I don't know of any references on essential singualrities. There be dragons. I know there are some refinements of C-W which replace a disk $D$ by a "slice of  pie", and that one can interpret functions with essential singularities as being those which do not extend to rational functions on the Riemann sphere, but I don't know how useful that is.
