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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544001.png" /> is an isolated [[Essential singular point|essential singular point]] of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544002.png" /> of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544003.png" />, then there exists at least one ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544004.png" /> issuing from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544005.png" /> such that in every angle
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If $a$ is an isolated [[Essential singular point|essential singular point]] of an analytic function $f(z)$ of the complex variable $z$, then there exists at least one ray $S=\{z:\arg(z-a)=\theta_0\}$ issuing from $a$ such that in every angle
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544006.png" /></td> </tr></table>
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$$V=\{z:|\arg(z-a)-\theta_0|<\epsilon\},\quad\epsilon>0,$$
  
that is symmetric with respect to the ray, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544007.png" /> assumes every finite value, except possibly one, at an infinite sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544008.png" /> converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j0544009.png" />. This result of G. Julia (see [[#References|[1]]]) supplements the big [[Picard theorem|Picard theorem]] on the behaviour of an analytic function in a neighbourhood of an essential singularity.
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that is symmetric with respect to the ray, $f(z)$ assumes every finite value, except possibly one, at an infinite sequence of points $\{z_k\}\subset V$ converging to $a$. This result of G. Julia (see [[#References|[1]]]) supplements the big [[Picard theorem|Picard theorem]] on the behaviour of an analytic function in a neighbourhood of an essential singularity.
  
The rays figuring in Julia's theorem are called Julia rays. Thus, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440011.png" />, the Julia rays are the positive and negative parts of the imaginary axis. In connection with Julia's theorem, a Julia segment or a Julia chord for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440012.png" /> meromorphic in, for example, the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440013.png" />, is a chord <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440014.png" /> with end point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440015.png" /> on the circumference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440016.png" /> such that in every open angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440017.png" /> with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440018.png" /> and containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440019.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440020.png" /> assumes all values on the Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440021.png" />-sphere, except possibly two. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440022.png" /> is called a Julia point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440023.png" /> if every chord <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440024.png" /> with end point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440025.png" /> is a Julia chord for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440026.png" />. There exist meromorphic functions of bounded characteristic for which every point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054400/j05440027.png" /> is a Julia point.
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The rays figuring in Julia's theorem are called Julia rays. Thus, for $f(z)=e^z$ and $a=\infty$, the Julia rays are the positive and negative parts of the imaginary axis. In connection with Julia's theorem, a Julia segment or a Julia chord for a function $w=f(z)$ meromorphic in, for example, the unit disc $D=\{z:|z|<1\}$, is a chord $S$ with end point $e^{i\theta_0}$ on the circumference $|z|=1$ such that in every open angle $V$ with vertex $e^{i\theta_0}$ and containing $S$ the function $w=f(z)$ assumes all values on the Riemann $w$-sphere, except possibly two. The point $e^{i\theta_0}$ is called a Julia point for $f(z)$ if every chord $S$ with end point $e^{i\theta_0}$ is a Julia chord for $f(z)$. There exist meromorphic functions of bounded characteristic for which every point on $|z|=1$ is a Julia point.
  
 
See also [[Asymptotic value|Asymptotic value]]; [[Iversen theorem|Iversen theorem]]; [[Cluster set|Cluster set]].
 
See also [[Asymptotic value|Asymptotic value]]; [[Iversen theorem|Iversen theorem]]; [[Cluster set|Cluster set]].

Latest revision as of 21:49, 22 December 2018

If $a$ is an isolated essential singular point of an analytic function $f(z)$ of the complex variable $z$, then there exists at least one ray $S=\{z:\arg(z-a)=\theta_0\}$ issuing from $a$ such that in every angle

$$V=\{z:|\arg(z-a)-\theta_0|<\epsilon\},\quad\epsilon>0,$$

that is symmetric with respect to the ray, $f(z)$ assumes every finite value, except possibly one, at an infinite sequence of points $\{z_k\}\subset V$ converging to $a$. This result of G. Julia (see [1]) supplements the big Picard theorem on the behaviour of an analytic function in a neighbourhood of an essential singularity.

The rays figuring in Julia's theorem are called Julia rays. Thus, for $f(z)=e^z$ and $a=\infty$, the Julia rays are the positive and negative parts of the imaginary axis. In connection with Julia's theorem, a Julia segment or a Julia chord for a function $w=f(z)$ meromorphic in, for example, the unit disc $D=\{z:|z|<1\}$, is a chord $S$ with end point $e^{i\theta_0}$ on the circumference $|z|=1$ such that in every open angle $V$ with vertex $e^{i\theta_0}$ and containing $S$ the function $w=f(z)$ assumes all values on the Riemann $w$-sphere, except possibly two. The point $e^{i\theta_0}$ is called a Julia point for $f(z)$ if every chord $S$ with end point $e^{i\theta_0}$ is a Julia chord for $f(z)$. There exist meromorphic functions of bounded characteristic for which every point on $|z|=1$ is a Julia point.

See also Asymptotic value; Iversen theorem; Cluster set.

References

[1] G. Julia, "Leçons sur les fonctions uniformes à une point singulier essentiel isolé" , Gauthier-Villars (1924)
[2] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. 345 (Translated from Russian)


Comments

Instead of Julia ray the term Julia direction is also used.

How to Cite This Entry:
Julia theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Julia_theorem&oldid=11784
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article