REMARKS ON ASYMPTOTIC EXPANSIONS OF COMPOSITIONS WITH THE ELEMENTARY FUNCTIONS  

求解初等函数复合的渐近展开补注

作　　者：LUO Xiao-Yu SHI Yong-Guo JIANG Zhi-Jie 罗小宇;石勇国;江治杰(四川轻化工大学数学与统计学院,四川宜宾644000;数据四川省恢复重点实验室,内江师范学院数学与信息学院,四川内江641100)

机构地区：[1]College of Mathematics and Statistics,Sichuan University of Science and Engineering,Sichuan 64300,China [2]Data Recovery Key Laboratory of Sichuan Province,College of Mathematics and Information Science,Neijiang Normal University,Sichuan 641100,China

出　　处：《数学杂志》2025年第2期131-139,共9页Journal of Mathematics

基　　金：Supported by The Innovation Fund of Postgraduate,Sichuan University of Science&Engineering(Y2024336);NSF of Sichuan Province(2023NSFSC0065).

摘　　要：In this paper,we study asymptotic power series of the composition f(x)=h(g(x)),where g(x)=∑_(n=0)^(∞)b_(n)x^(-n),b_(n)∈R,and h is a given elementary function.The asymptotic expansions have been obtained for the composition with an exponential or logarithmic function.Using the re-cursive method,we present the asymptotic expansions for the composition with seven trigonometric functions,respectively.As an application,the asymptotic expansions of roots of some equations are given.Computational results show that our recursive formula is more efficient than the method of Lagrange's inverse theorem.本文研究了复合函数f(x)=h(g(x))的渐近幂级数,其中g(x)=∑_(n=0)^(∞)b_(n)x^(-n),b_(n)∈R,h为给定初等函数.当h是指数或对数函数时,复合函数的渐近展开已有结果.利用递推法,本文分别获得了七个三角函数复合的渐近展开式.作为应用,还给出了方程根的渐近展开.计算结果显示,我们的递推公式比拉格朗日逆定理的方法更有效率.

关 键 词：asymptotic expansion asymptotic power series trigonometric function composite function 

分 类 号：O173.1[理学—数学] O241.5[理学—基础数学]