Borel的一个定理的一个应用——纪念刘书琴先生逝世一周年

AN APPLICATION OF A THEOREM OF BOREL

在这篇论文中,研究了一般形式的代数微分方程的亚纯解的增长性并得到一些结果,研究的方法是根据一个关于亚纯函数组的定理。这个定理是Borel的一个关于整函数组的定理的一个推广。

In a paper of 1897, Borel showed that if two systems of entire functions Gi(z) (i = 1, 2, … , n) and Hi(z) (i = 1,2,… ,n) satisfy the identityand if the growth of Gi(z) (1≤ i≤ n) is, in a certain sense, slower than that of eHj(z)-Hk(z) (1 j, k n, j k), thenThis theorem of Borel is extended to the case of two systems of meromorphic functions fj (z) (j=1,2, … ,n) and Gj(z) (j=1,2, … ,n) by Chuang in 1991 who proved that, under certain conditions, the identityimpliesfj(z)≡ 0 (j=1,2, … ,n).The main purpose of the present paper is to apply the latter theorem to study the growth of meromorphic solutions of algebraic differential equation in general form. Some results are obtained.