Behavior of essential singularities in an 'open cone'

Question

Picard's Big Theorem says that if a function $f(z)$ has an isolated essential singularity at a point $w$, then in every neighborhood of $w$, $f(z)$ hits every complex number infinitely many times, with perhaps at most one exception.

Is there a version of Picard's theorem that goes something like this?

Let $V$ be an open disc (finite radius) such that $f(z)$ is holomorphic on $V - \lbrace w \rbrace$, and has an essential singularity at $w$. Let $0 \leq \theta < \phi < 2\pi$, and define $Cone(w,V,\theta,\phi)$ to be $V \cap \lbrace w + re^{i\varphi} \mid r > 0, \theta < \varphi < \phi \rbrace$. Think of this as a "pizza slice" of the disc $V$.

Is it true that there exists an $\alpha$ such that $f(z) = \alpha$ for infinitely many $z\in Cone(w,V,\theta,\phi)$?

Answer 1

Maybe it's not exactly what you are asking for (and maybe you know it already), but a related concept to what you are asking is that of Julia line.

For sake of simplicity, consider an entire function $f$ with an essential singularity at $\infty$; let
$S(\phi,\epsilon)=\{z\ :\ |\mathrm{arg}(z)-\phi|<\epsilon\}$
be a sector around the line $R(\phi)=\{re^{i\phi}\ :\ r\geq0\}$. We call $R(\phi)$ a Julia line if, for every $\epsilon>0$, $f$ takes on every complex value in $S(\phi,\epsilon)$ with possibly one exception infinitely many times.

$R(\phi)$ is a weak Julia line if for every $\epsilon>0$, every $r>0$, the image of $S(\phi,\epsilon)\cap\{|z|>r\}$ is dense in the complex plane.

Both notions are stronger than what you are asking for and both deal with entire functions rather than local behaviour around isolated essential singularities, but it could be a starting point.

Results

• Every trascendental function has a weak Julia line (I don't have any reference for this, but it is more an exercise in one complex variables)
• Every trascendental function has a Julia line (you can look up in Cartwright, Integral functions)
• If $f(z)=\sum a_k z^{n_k}$ and $n_k/k\to \infty$, then $f$ takes on every complex value infinitely many times in every $S(\phi,\epsilon)$ (Hayman, Angular value distributions of power series with gaps)

Another reference I know about this stuff is Anderson, Clunie, Entire functions of finite order and lines of Julia.

Warning -- I don't know of any example of a weak Julia line which isn't a Julia line, so the two concepts could very well coincide. But I think it is an open problem.

Answer 2

Consider $\exp(1/z)$ in the region $\Re(z)>0$. This has an essential singularity at zero, but its image is the complement of the unit ball.