Behavior of essential singularities in an 'open cone' - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:21:22Z http://mathoverflow.net/feeds/question/72102 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72102/behavior-of-essential-singularities-in-an-open-cone Behavior of essential singularities in an 'open cone' Henry Yuen 2011-08-04T17:30:30Z 2011-08-12T17:04:43Z <p><a href="http://en.wikipedia.org/wiki/Picard_theorem" rel="nofollow">Picard's Big Theorem</a> 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.</p> <p>Is there a version of Picard's theorem that goes something like this?</p> <p>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 $0 \leq \theta &lt; \phi &lt; 2\pi$, and define $Cone(w,V,\theta,\phi)$ to be $V \cap \lbrace w + re^{i\varphi} \mid r > 0, \theta &lt; \varphi &lt; \phi \rbrace$. Think of this as a "pizza slice" of the disc $V$.</p> <p>Is it true that there exists an $\alpha$ such that $f(z) = \alpha$ for infinitely many $z\in Cone(w,V,\theta,\phi)$?</p> http://mathoverflow.net/questions/72102/behavior-of-essential-singularities-in-an-open-cone/72104#72104 Answer by Colin McQuillan for Behavior of essential singularities in an 'open cone' Colin McQuillan 2011-08-04T17:48:35Z 2011-08-04T17:48:35Z <p>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.</p> http://mathoverflow.net/questions/72102/behavior-of-essential-singularities-in-an-open-cone/72774#72774 Answer by Samuele for Behavior of essential singularities in an 'open cone' Samuele 2011-08-12T14:48:50Z 2011-08-12T17:04:43Z <p>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 <em>Julia line</em>.</p> <p>For sake of simplicity, consider an entire function $f$ with an essential singularity at $\infty$; let<br> $S(\phi,\epsilon)=\{z\ :\ |\mathrm{arg}(z)-\phi|&lt;\epsilon\}$<br> be a sector around the line $R(\phi)=\{re^{i\phi}\ :\ r\geq0\}$. We call $R(\phi)$ a <em>Julia line</em> if, for every $\epsilon>0$, $f$ takes on every complex value in $S(\phi,\epsilon)$ with possibly one exception infinitely many times. </p> <p>$R(\phi)$ is a <em>weak Julia line</em> if for every $\epsilon>0$, every $r>0$, the image of $S(\phi,\epsilon)\cap\{|z|>r\}$ is dense in the complex plane.</p> <p>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. </p> <p><strong>Results</strong></p> <ul> <li>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)</li> <li>Every trascendental function has a Julia line (you can look up in Cartwright, <em>Integral functions</em>)</li> <li>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, <em>Angular value distributions of power series with gaps</em>) </li> </ul> <p>Another reference I know about this stuff is Anderson, Clunie, <em>Entire functions of finite order and lines of Julia</em>.</p> <p><strong>Warning</strong> -- 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.</p>