The Value Distribution Theory And Its Application Research

Using the Nevanlinna value distribution theory of meromorphic function and some skills of differential equations theory, this taper maily investigates the problems of the value distribution of differential polynomials, and the existence and the order of meromo-rphic solutions of some systems of functional equations or algebraic differential equations . This taper is divided into five chapters:In chapter one, we desribe the basic Nevanlinna theory of meromorphic function and algebroid function, including fundamental definitions and fundamental theorems.In chapter two, using the Nevanlinna value distribution theory of meromorphic function, investigate the problems of the value distribution of differential polynomials, improve some previous results, and examples show that our conclusions are better.In chapter three, using the Nevanlinna value distribution theory of meromorphic function, we conserder the problems of the existence and the order of meromorphic solutions of some systems of functional equations, and generalize some results of GGundersen etc.In chapter four, using the Nevanlinna value distribution theory of meromorphic function and algebroid function theory, we discuss some properties of the transcendental meromorphic solutions of second-order algebraic differential equations.In chapter five, by Nevanlinna theory of the value distribution of meromorphic functions and the theory of the uniqueness of meromorphic functions, some theorems on systems of meromorphic functions are investigated, some results of professor Yi Hongxun are generalized, and the results obtained here are the theorems on systems of meromorphic functions involving small functions.