Uniqueness Of The Meromorphic Functions And The Normal Family

Nevanlinna Theory(refer to[1],[2],[3])as one of the most perfect achievements in last century not only takes an important position in mathematics,but also inherits the great value in applications from function analysis.Nevanlinna Theory paves the path of research on value distribution of meromorphic functions,constructs characteristic function and overcomes the restriction of meaningless of maximum module function on meromorphie functions.Due to the broad correlation with other fields and mutual influcnce,both Nevanlinna Theory and others take big strides. The value distribution of module of function directly based on Nevanlinna Theory provides a new approach in research on differential equation.It gives a decent example for the angular value distribution theory and value distribution theory on p-adic field,n-dimentional complex Euclidian space,and even on general manifold.Morcover,Nevanlinna's value distribution theory of meromorphic functions has been used to study or tackle the growth,oscillation,solvabilily and existence of entire or meromorphic solutions of differential equations in complex domains.The value distribution theory estanbished by Nevanlinna becomes naturally the main tool for the studies.Early,Nevanlinna proved that any nonconstant meromorphic function can be uniquely determined by five values.Certainly.the number of five can be reduced when more conditions been added.This paper is concerned with the unicity of the meromorphic functions which share sets or values and a proof about the normal family.In Chapter 1,we describe the basic Nevanlinna Theory briefly.In Chapter 2,we will give a therom about the unicity of meromorphic which share a set and satisfy some relation of deficient values.Chapter 3 proves a therom about normal family.