On W. Gordon's integral (1929) and related identities

TL;DR

This paper analytically evaluates W. Gordon's integral from 1929, providing convergence conditions and demonstrating its broad applicability in deriving numerous definite integrals relevant to theoretical physics.

Contribution

It introduces a comprehensive analytic evaluation of Gordon's integral, expanding its utility in mathematical physics and related fields.

Findings

Derived explicit formulas for Gordon's integral

Established convergence conditions for the integral

Showcased applications in physics-related integral evaluations

Abstract

Analytic evaluation of Gordon's integral $$\operatorname {J}_c^{j(\pm p)}(b,b';\lambda,w,z)=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(b';c\pm p;zx)dx,$$ are given along with convergence conditions. It shows enormous number of definite integrals, frequently appear in theoretical and mathematical physics applications, easily deduced from this generalized integral.