A table of Legendre-transformation pairs with methodologies for construction, authentication, and approximation of pairs

TL;DR

This paper provides an extensive table of Legendre transformation pairs, including methods for constructing, authenticating, and approximating these pairs, with applications to special functions and formulas in sciences.

Contribution

It introduces a comprehensive table of Legendre pairs along with methodologies for their construction, verification, and approximation, covering analytic functions and polynomial approximations.

Findings

Includes many special functions and formulas used in sciences.

Provides formulations for Legendre transformations of analytic functions.

Offers methods for polynomial approximation of functions.

Abstract

An extensive table of pairs of functions linked by the Legendre transformation is presented. Many special functions and formulas that are used in the sciences are included in the pairs. Formulations are provided for finding the Legendre transformations of analytic functions and of polynomial approximations to other functions.