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

Unified Analytic Solution of Transcendental Equations Involving 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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 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.176132895.57340960/v1 126 views 122 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统一微分代数框架专业化并扩展到涉及双曲函数的重要方程类别。我们严格构造了任何截断阶数 M 的先验微分代数闭包 K （M） trans ，显式地与双曲正弦和余弦函数以及它们定义的微分关系和代数恒等式相邻。我们证明相应截断方程 fM （x） = 0 的根可以在该闭包内解析表达。该解公式推广了已建立的框架，结合了临界点分析、泰勒级数截断、组合校正项和谱分解，现在适用于双曲函数。我们提供了建设性证明，推导出校正系数γ （M） m 的显式组合表达式，并提出了详细的 O（M 2） 算法。对代表性双曲方程的数值验证显示出高精度的准确性，截断方程的残差通常低于 10 −28，并且与原始根呈指数收敛。这项工作确立了微分代数闭包方法对工程和物理学中普遍存在的一类基本超越方程的实际适用性。 Supplementary Material File (hyperbolic1 (1).pdf) Download 435.32 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 hyperbolic functions numerical computation 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 126 views 122 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Unified Analytic Solution of Transcendental Equations Involving Hyperbolic Functions in Differential Algebraic Closure. Authorea . 24 October 2025. DOI: https://doi.org/10.22541/au.176132895.57340960/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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