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!

  • $\begingroup$ First of all I think it would help to clear up terminology: Meromorphic functions can have infinitely many poles. You are talking about funcitons holomorphic except for poles, so it is confusing to say "need not be holomorphic (or meromorphic)". $\endgroup$ – Jonas Meyer Jun 30 '11 at 6:17
  • $\begingroup$ Your argument is incorrect where you claim that you can choose a punctured neighborhood of $z_0$ where $f$ is nonzero. The theorem you applied would only be valid if $f$ were holomorphic in a domain containing $z_0$. For example, consider $\tan\left(\frac{1}{z}\right)$ with $z_0=0$. $\endgroup$ – Jonas Meyer Jun 30 '11 at 6:24
  • $\begingroup$ Could you please make this post more self-contained by putting the relevant mathematics in the question (not the whole thing, but enough)? Besides the inconvenience of having to click through, it would keep a record of what is being asked here, rather than in a document that might be altered or change address. $\endgroup$ – Jonas Meyer Jun 30 '11 at 6:33

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.

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