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!