{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T07:03:10Z","timestamp":1769756590410,"version":"3.49.0"},"reference-count":51,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T00:00:00Z","timestamp":1753315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We introduce a modern methodology for constructing global analytical approximations of special functions over their entire domains. By integrating the traditional method of matching asymptotic expansions\u2014enhanced with Pad\u00e9 approximants\u2014with differential evolution optimization, a modern machine learning technique, we achieve high-accuracy approximations using elegantly simple expressions. This method transforms non-elementary functions, which lack closed-form expressions and are often defined by integrals or infinite series, into simple analytical forms. This transformation enables deeper qualitative analysis and offers an efficient alternative to existing computational techniques. We demonstrate the effectiveness of our method by deriving an analytical expression for the Fermi gas pressure that has not been previously reported. Additionally, we apply our approach to the one-loop correction in thermal field theory, the synchrotron functions, common Fermi\u2013Dirac integrals, and the error function, showcasing superior range and accuracy over prior studies.<\/jats:p>","DOI":"10.3390\/axioms14080566","type":"journal-article","created":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T14:11:44Z","timestamp":1753366304000},"page":"566","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Analytical Approximations as Close as Desired to Special Functions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3620-0878","authenticated-orcid":false,"given":"Aviv","family":"Orly","sequence":"first","affiliation":[{"name":"School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,24]]},"reference":[{"key":"ref_1","unstructured":"Geng, Z., Abdulah, S., Sun, Y., Ltaief, H., Keyes, D.E., and Genton, M.G. 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