Unified Analytic Solution of Transcendental Equations Involving Inverse Hyperbolic Functions in Differential Algebraic Closure

Unified Analytic Solution of Transcendental Equations Involving Inverse Hyperbolic Functions in Differential Algebraic Closure | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 24 October 2025 V1 Latest version Share on Unified Analytic Solution of Transcendental Equations Involving Inverse Hyperbolic Functions in Differential Algebraic Closure Authors : Dongqi Liu 0009-0006-4018-9292 and shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176132896.60866401/v1 121 views 177 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文将求解解析超越方程的严格微分代数框架扩展到涉及反双曲函数的特定类方程。我们为任何截断阶数 M 构建了一个定义明确的超越微分代数闭包 K （M） trans，并证明相应截断方程 fM （x） = 0 的根，其中 f 包含 arsinh（x）、arcosh（x） 或 artanh（x），可以在该闭包内解析表达。解公式通过临界点分析、泰勒级数截断、组合校正项和谱分解对多项式情况进行了推广，并针对逆双曲函数的微分性质进行了具体调整。我们提供了具有完整数学细节的建设性证明，通过严格的推导推导了校正系数γ （M） m 的组合表达式，并提出了具有全面稳定性分析的详细 O（M 2） 算法。对代表性方程的广泛数值验证证明了高精度的准确性，截断方程的残差通常低于 10 −28。误差传播分析以及与传统方法的比较确定了我们方法的鲁棒性和效率。这项工作通过证明显式解析解存在于适当扩展的微分代数闭包 K （M） trans 中，为求解这些非基本方程提供了一个统一的框架，从而与经典的不可能结果相协调。 Supplementary Material File (inverse_hyperbolic.pdf) Download 512.00 KB Information & Authors Information Version history V1 Version 1 24 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords analytic solution combinatorial correction critical point analysis differential algebraic closure inverse hyperbolic functions numerical computation taylor truncation transcendental equations Authors Affiliations Dongqi Liu 0009-0006-4018-9292 View all articles by this author shifa liu 0009-0003-6570-2812 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 121 views 177 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Unified Analytic Solution of Transcendental Equations Involving Inverse Hyperbolic Functions in Differential Algebraic Closure. Authorea . 24 October 2025. DOI: https://doi.org/10.22541/au.176132896.60866401/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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