初等函数是由基本初等函数经过有限次的四则运算和复合运算所得到的函数。基本初等函数和初等函数在其定义区间内均为连续函数。不是初等函数的函数，称为非初等函数，如狄利克雷函数和黎曼函数。目前有两种分类方法：数学分析有六种基本初等函数，高等数学只有五种。
第三种方法是把待求级数用基本初等函数的幂级数展开式表示出来；
The third one is to let series be expressed by basic elementary function's power series expansion;
用求解常微分方程及其初值问题的方法得到由函数方程表示的基本初等函数。
The basic elementary functions represented by functional equations were obtained by using the methods of solving ordinary differential equations and initial value probe.
运用微积分的方法，对基本初等函数的若干特征性质进行了证明。从而给出了基本初等函数的等价表征。
This paper, using the method of differential equation, proves some characters properties of basic elementary function, and then concludes an equivalent representation about basic elementary function.
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