外文翻译--些周期性的二阶线性微分方程解的方法 2.doc

Some Properties of Solutions of Periodic Second Order Linear Differential Equations Introduction and main results In this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinnas value distribution theory of meromorphic functions [12, 14, 16]. In addition, we will use the notation，and to denote respectively the order of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function ，（[see 8]），the e-type order of f(z), is defined to be Similarly, ，the e-type exponent of convergence of the zeros of meromorphic function , is defined to be We say thathas regular order of growth if a meromorphic functionsatisfies We consider the second order linear differential equation Where is a periodic entire function with period . The complex oscillation theory of (1.1) was first investigated by Bank and Laine [6]. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained [2{11, 13, 17{19]. Whenis rational in ，Bank and Laine [6] proved the following theorem Theorem A Letbe a periodic entire function with period and rational in .Ifhas poles of odd order at both and , then for every solutionof (1.1), Bank [5] generalized this result: The above conclusion still holds if we just suppose that both and are poles of, and at least one is of odd order. In addition, the stronger conclusion (1.2) holds. Whenis transcendental in, Gao [10] proved the following theorem Theorem B Let ,whereis a transcendental entire function with, is an odd positive integer and，Let .Then any non-trivia solution of (1.1) must have. In fact, the stronger conclusion (1.2) holds. An example was given in [10] showing that Theorem B does not hold when is any positive integer. If the order , but is not a positive integer, what can we say? Chiang and Gao [8] obtained the following theorems Theorem 1 Let ,where,andare entire funct