关于解析函数的奇异点

关于解析函数的奇异点 ON SINGULARITIES OF ANALYTIC FUNCTIONS 许全华 Xu Quanhua first-author 本文利用Mandelbrojt[1,2]和Valiron[3]的方法给出了Taylor级数所表示的解析函数以其收敛圆上某点为奇异点的两个判别准则。其次,利用Valiron[3]关于奇异点的幅角公式推广了Mandelbrojt[2]关于缺项Taylor级数在其收敛圆周上的某段弧上一定具有奇异点的定理。最后,结合所得结果和Agmon[6]关于Dirichlet级数的奇异点的复合定理,给出了某些Dirichlet级数在其解析轴上的奇异点的位置。 In this paper we consider singularities of analytic functions represented by power series. Let f（z） be an analytic function represented by Taylor series: f（z）=（1） whose radius of convergence is unit. The main results of this paper as follows: Theorem 1 Let f（z） be defined by the series （1） whose radius of convergence is equal to 1. Then a necessary and sufficient condition for z=1 to be a singular point of f（z） is that there erists a positive number h > 0 such that D（h）=1+h, where or equivalently, that there exists an h∈（0,1） such that R（h）=1-h, where Theorem 3 Let the series （1） have 1 as its radius of convergence) and let {μ j } be a sequence of positive numbers converging to zero as j→∞. If there exists a non-negative constant a, and if to each j corresponds a sequence {m n j } of positive integers such that ⅱ) a m =0, me[（1-μj）m n j ,m n j ), then the function f（z） defined by the series （1） has at least a singular point on each arc of |z|=1 of length 2|cos -1 e -a | (|cos -1 e -a | < π/2). Thoorem 4 Let f（z）, {μ j } and a be defined as theorem 3. If to each j correspons a sequence {m n j } of positive integers and a sequence {β n j } of real numbers such that then there exists at least a singular point of f（z） on the are of |z| =1 with z=1 as its center and length 2 (cos -1 e -a |(|cos -a e -a | < π/2). 1985-03-01 2021-04-01 3