Analytical approximations as close as desired to special functions

TL;DR

This paper introduces a method to create highly accurate, globally valid analytical approximations for complex functions, facilitating easier analysis and evaluation especially when closed-form solutions are unavailable.

Contribution

The authors develop a novel approach to construct analytical expressions that closely approximate special functions across their entire domain, including those defined by integrals or series.

Findings

Provides accurate approximations for quantum gas pressure and density

Replaces complex functions with simple analytical expressions

Enables qualitative analysis and simplifies evaluations

Abstract

We present a method for constructing global analytical expressions that approximate a function over its entire range. These approximations not only mirror the original function as accurately as desired, but are purposefully created to possess features that the original function lacks. This is particularly useful for functions that lack closed form and are defined by integrals or infinite series. Replacing these definitions with simple analytical expressions enables in-depth qualitative analysis and replaces the current methods of evaluation. We demonstrate this procedure by providing replacements for a variety of pivotal functions in physics and cosmology including the pressure and density of quantum gas, the one-loop correction in thermal field theory, common polylog functions, and the error function.