Analytical Approximations as Close as Desired to Special Functions

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—enhanced with Padé approximants—with 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–Dirac integrals, and the error function, showcasing superior range and accuracy over prior studies.