高阶线性微分方程的解与小函数的关系研究

The Relation between the Solutions of Higher Order Linear Differential Equations and Small Functions

利用值分布理论研究复微分方程解的增长级及解与小函数的关系.在复数域内分别定义数目函数、平均中值函数及特征函数,运用Poisson-Jesen公式建立值分布理论的基本定理,将所定义的函数引入Jesen公式,变换后获得值分布理论的第一基本定理和第二基本定理.在复平面上考虑高阶微分方程,研究方程系数[p,q]解的增长级问题.对于高阶非齐次线性微分方程,方程解多项式级为小于n的亚纯函数,取小函数上点时收敛指数为无穷大.

The relation between the growth stage and the solution of the complex differential equation and the small function is studied by the value distribution theory.In the complex domain,the number function,the mean median function and the characteristic function are defined respectively,and the basic theorem of the value distribution theory is established by using the Poisson-Jesen formula.The defined function is introduced into the Jesen formula.After transformation,the first fundamental theorem and the second fundamental theorem of the value distribution theory are obtained.In the complex plane,the higher order differential equations are considered.Study on the coefficients of equation,the growth of solutions of[p,q].For higher order nonhomogeneous linear differential equations,solution of polynomial is less than n of meromorphic function equation,taking small function point convergence index is infinite.