Julia theorem

If is an isolated essential singular point of an analytic function of the complex variable , then there exists at least one ray issuing from such that in every angle

that is symmetric with respect to the ray, assumes every finite value, except possibly one, at an infinite sequence of points converging to . 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 and , 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 meromorphic in, for example, the unit disc , is a chord with end point on the circumference such that in every open angle with vertex and containing the function assumes all values on the Riemann -sphere, except possibly two. The point is called a Julia point for if every chord with end point is a Julia chord for . There exist meromorphic functions of bounded characteristic for which every point on 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.