{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-limits-of-a-family-of-asymptotic","title":"The Limits of a Family; of Asymptotic Solutions to The Tetration Equation","arxiv_id":"2104.01990","date":"2021-04-01","proceeding":null,"authors":["James David Nixon"],"abstract":"In this paper we construct a family of holomorphic functions $\\beta_\\lambda (s)$ which are solutions to the asymptotic tetration equation. Each $\\beta_\\lambda$ satisfies the functional relationship ${\\displaystyle \\beta_\\lambda(s+1) = \\frac{e^{\\beta_\\lambda(s)}}{e^{-\\lambda s} + 1}}$; which asymptotically converges as $\\log \\beta_\\lambda(s+1) = \\beta_\\lambda (s) + \\mathcal{O}(e^{-\\lambda s})$ as $\\Re(\\lambda s) \\to \\infty$. This family of asymptotic solutions is used to construct a holomorphic function $\\text{tet}_\\beta(s) : \\mathbb{C}/(-\\infty,-2] \\to \\mathbb{C}$ such that $\\text{tet}_\\beta(s+1) = e^{\\text{tet}_\\beta(s)}$ and $\\text{tet}_\\beta : (-2,\\infty) \\to \\mathbb{R}$ bijectively.","url_abs":"https://arxiv.org/abs/2104.01990v4","url_pdf":"https://arxiv.org/pdf/2104.01990v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-limits-of-a-family-of-asymptotic","repo_url":"https://github.com/JmsNxn92/Recursive_Tetration_PARI","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}