Some Uniqueness Theories About Meromorphic Functions And Algebroid Functions

In 1929, Nevanlinna proved his famous four-value (4CM) theorem which is an important result in uniqueness theory of meromorphic functions. In 1979, Gundersen proved a further result and generalized it to 2CM+2IM=4CM. There is example to show that 4IM≠4CM, but the question that 1CM+3IM=?4CM is still open. In 2000, Lahira gave a method--weighted sharing to research sharing values of meromorphic functions and used this method to improve some results of meromorphic functions.In the late of 1950's professor Xiong Qing-Lai and his students did some researches in combination between Meromorphic functions and Nevanlinna theory of Algebroid functions. The professors Sun Dao-Chun and Gao Deng-Sheng researched the plus of algebroid functions and generalized the results of the uniqueness theory of meromorphic functions to algebroid functions.In this paper, I will introduce the following contents:The first chapter introduces some basic results in Nevanlinna theory including the definitions of m ( r , f ), N ( r , f ), T ( r , f ), and the first fundamental theorem and the second fundamental theorem of meromorphic functions.The second chapter introduces some knowledge about algebroid functions including the first fundamental theory and the second fundamental theory of algebroid functions. At the same time introduces a result in algebroid function which I got by extending a result of meromorphic functions to algebroid functions.The third chapter will give the definition of weighted sharing and some results in four-value theorem including a result which I got.