复平面和单位圆内高阶线性微分方程解同小函数的关系
发布时间：2018-09-08 13:54
【摘要】：本文运用Nevanlinna值分布理论研究了系数为亚纯函数及系数为周期函数的几类高阶线性微分方程解的复振荡性质.全文分为四章.第一章,简要概述了复线性微分方程领域的发展历史,并介绍了复平面和单位圆内的解析函数和亚纯函数的一些基本概念和常用记号.第二章,研究了复平面上高阶线性微分方程亚纯解及其一阶导数同小函数的关系问题,得到了微分方程解取小函数的点的收敛指数.第三章,研究了复平面上一类高阶线性周期微分方程非平凡次正规解的存在性和表示形式,得到了高阶线性周期微分方程解的一些性质.第四章,研究了在一定条件下单位圆内高阶线性微分方程亚纯解的增长性,并对微分方程解取小函数时的收敛指数进行了估计。
[Abstract]:In this paper, we study the complex oscillation properties of solutions of higher order linear differential equations whose coefficients are meromorphic functions and whose coefficients are periodic functions by using the Nevanlinna value distribution theory. The full text is divided into four chapters. In the first chapter, the development history of complex linear differential equations is briefly summarized, and some basic concepts and common notations of analytic and meromorphic functions in complex plane and unit circle are introduced. In chapter 2, we study the relationship between meromorphic solutions of higher order linear differential equations and their first order derivatives and small functions on complex plane, and obtain the convergence exponents of the points in which the solutions of differential equations take small functions. In chapter 3, we study the existence and representation of nontrivial subnormal solutions for a class of higher order linear periodic differential equations on complex plane, and obtain some properties of solutions of higher order linear periodic differential equations. In chapter 4, we study the growth of meromorphic solutions of higher order linear differential equations in a unit circle under certain conditions, and estimate the convergence exponents when the solutions of differential equations are small.
【学位授予单位】：江西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.5
本文编号：2230715
[Abstract]:In this paper, we study the complex oscillation properties of solutions of higher order linear differential equations whose coefficients are meromorphic functions and whose coefficients are periodic functions by using the Nevanlinna value distribution theory. The full text is divided into four chapters. In the first chapter, the development history of complex linear differential equations is briefly summarized, and some basic concepts and common notations of analytic and meromorphic functions in complex plane and unit circle are introduced. In chapter 2, we study the relationship between meromorphic solutions of higher order linear differential equations and their first order derivatives and small functions on complex plane, and obtain the convergence exponents of the points in which the solutions of differential equations take small functions. In chapter 3, we study the existence and representation of nontrivial subnormal solutions for a class of higher order linear periodic differential equations on complex plane, and obtain some properties of solutions of higher order linear periodic differential equations. In chapter 4, we study the growth of meromorphic solutions of higher order linear differential equations in a unit circle under certain conditions, and estimate the convergence exponents when the solutions of differential equations are small.
【学位授予单位】：江西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.5
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