Julia theorem

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.