数学论文英文翻译中英-可控的无穷时滞中立型泛函微分方程.doc

英文翻译 下属学院 理工学院 专 业 信息与计算科学 班 级 08信息与计算科学 学 号 2008009010 姓　 名 指导教师 　　 2011 年 11 月 25日 文献翻译一：原文 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 an