A Lagrangian Method for Deriving New Indefinite Integrals of Special Functions

TL;DR

This paper introduces a Lagrangian-based method to derive indefinite integrals of special functions, producing many new and known integrals by solving second-order linear differential equations.

Contribution

It presents a novel Lagrangian formulation approach to generate indefinite integrals of special functions, expanding the set of known integrals and enabling new combinations of functions.

Findings

Derived new indefinite integrals for Bessel, Airy, Legendre, and hypergeometric functions.

Produced extensive results for elliptic integrals of the first and second kinds.

Demonstrated the method's ability to generate integrals involving multiple special functions.

Abstract

A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the method. Sample results are given for Bessel functions, Airy functions, Legendre functions and hypergeometric functions. More extensive results are given for the complete elliptic integrals of the first and second kinds. Integrals can be derived which combine common special functions as separate factors.