Characteristics of a Meromorphic Function

We characterize meromorphic functions in terms of points at which they assume some values. The purpose is realized by using their characteristics. In this chapter, we introduce the Nevanlinna’s characteristic in a domain (especially in a disk centered at the origin), the Nevanlinna’s characteristic in an angle and the Tsuji’s characteristic in terms of the generalized Poisson formula, Carleman formula and Levin formula respectively. These formulae are derived from the second Green formula. Similarly, the introduction of the Ahlfors-Shimizu characteristic originates from the second Green formula from the point of view of analysis. We exhibit the first and second fundamental theorems for every type of characteristics and the estimates of error terms, especially that of corresponding error terms to the Nevanlinna’s characteristic in an angle. The relationship among various characteristics and among the integrated counting functions are found out. These relationship make us to produce new results and applications. We compare the characteristics of meromorphic functions and their derivatives. Next in terms of the Ahlfors-Shimizu’s characteristic for an angle, we make a careful discussion of value distribution of functions meromorphic in an angle, especially theorems of the Borel-type. Then we discuss deficiency and deficient values which includes an introduction to Baernstein’s spread relation along with related results. Finally, we establish a series of unique theorems of meromorphic functions in an angle in terms of Tsuji’s characteristic. This is a new topic.