T Directions of a Meromorphic Function

Abstract

A transcendental meromorphic function has a singular property in any neighborhood of its essential singular point, for example, it assumes there infinitely often all but at most two values on the extended complex plane. This property is preserved in any angular domain containing some fixed ray. Such ray is termed as the singular direction of the function considered. In this chapter, we mainly discuss T directions of meromorphic functions, which was introduced by the author in 2003. First of all we consider the existence of T directions including T directions with small functions as targets. Next we consider connections among T directions and other directions such as Julia directions and Borel directions and mainly introduce a result of Zhang Q. D. which proves that a T (resp. Borel) direction may not be a Borel (resp. T) direction. We list conditions for the existence of singular directions dealing with derivatives of the functions, that is, the Hayman T directions and for the existence of common T directions of a function and its derivatives. We present a simple discussion of distribution of the Julia, Borel and T directions. In terms of their asymptotic form, through the Stokes rays we investigate singular directions of meromorphic solutions of a linear differential equation with rational coefficients. In the case of at least one of the coefficients being transcendental, we use the Nevanlinna’s fundamental theorems for an angle to attain the aim of our researches. We conclude this chapter with a simple survey on value distribution of algebroid functions including the Nevanlinna first and second fundamental theorems for a disk and unique theorems and the singular directions.