一类幂指函数列的分析性质

The Properties of a Sequence of Exponential Functions

讨论幂指函数列{yn(x)}(n≥0,x∈[0,1])的一些分析性质,研究这些函数的几何图形分布情况,其中y0(x)=1,yn(x)=xyn-1(x)(n≥1)且y2k(0)=1,y2k+1(0)=1(k≥0).可获知函数列{yn(x)}中的每一个函数yn(x)均在闭区间[0,1]上一致连续,在开区间(0,1)内可导,且在闭区间[0,1]上一致收敛于连续的和函数y=y(x),其中y=y(x)由关系式y=xy(0<x,y<1))惟一确定,并且y(0)=0,y(1)=1.

This paper studies the properties of the exponential functions {y.（x）} on [0,1] defined by yo （x） = 1, yn（x） x^yn-1（x）forn=1,2,…, andy2k（0）=1, Y2k＋l（0）=l（k^0）. The result shows that each function yn（x） is continuous over [0,1], and is differentiable on the open lnterva lnterva （0,1）. Moreover, the sequence uniformly converges to a function y = y（x） in the closed [0,1], which is defined by y = xy（0 〈x,y〈1）.