数学论文 英文翻译.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
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 Nevanlinna's 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 . plex oscillation theory of () was first investigated by Bank and Laine [6]. Studies concerning () 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 (),
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
()
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 () must have. In fact, the stronger conclusion () 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 th