Meromorphic Function With Derivatives Sharing Common Values Or Small Function

In 1925, R.Nevanlinna established the value distribution theory of meromorphic function on complex plane C.That was the started sign about uniqueness theory of meromorphic function in modern times.In this paper,the uniqueness for sharing common values and common small function of k order derivatives of meromorphic functions is deeply discussed.By the use of auxiliary function and the calculation of regrade zeros or regrade poles, some results about the uniqueness of meromorphic function were obtained. And the whole paper is divided into three chapters.The first chapter is about preliminary theory. We briefly introduced the de-velopment of Nevanlinna's theory and some basic theorems on Nevanlinna's theory and uniqueness theory of meromorphic function.In the second chapter, we discussed the uniqueness problems on sharing com-mon value of k order derivatives of two meromorphic function. In different methods, four relative results were obtained.In the third chapter, the discussing was continued. We investigated the unique-ness problems on sharing common small function of k order derivatives of two mero-morphic function. And we got four relative results, which improved the results given by G.D.Qiu, P.Li, W.J.Yuan.