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| 摘要: |
| 本文研究了复合函数$f(x)=h(g(x))$ 的渐近幂级数, 其中$g(x) = \sum_{n=0}^{\infty} b_n x^{-n}$, $b_n\in \mathbb{R}$, $h$ 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: 渐近展开 渐近幂级数 三角函数 复合函数 |
| DOI： |
| 分类号:O173.1 ; O241.5 |
| 基金项目:四川轻化工大学研究生创新基金资助项目(Y2024336);省部级科研项目（2023NSFSC0065） |
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| Remarks on asymptotic expansions of compositions with the elementary functions |
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Luo xiaoyu,Shi Yong-Guo,江治杰
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| Abstract: |
| In this paper, we study asymptotic power series of the composition $f(x)=h(g(x))$, where $g(x) = \sum_{n=0}^{\infty} b_n x^{-n}$, $b_n\in \mathbb{R}$, and $h$ is a given elementary function. The asymptotic expansions have been obtained for the composition with an exponential or logarithmic function. Using the recursive method, we present the asymptotic expansions for the composition with seven trigonometric functions, respectively. As an application, the asymptotic expansions of roots of some equation are given. Computational results show that our recursive formula is more efficient than the method of Lagrange's inverse theorem. |
| Key words: asymptotic expansion asymptotic power series trigonometric function composite function |
