Integrals of Lipschitz-Hankel Type, Legendre Functions, and Table Errata

TL;DR

This paper reviews the evaluation of Lipschitz-Hankel integrals involving Legendre functions, corrects errors in standard references, and discusses their applications in communication theory.

Contribution

It provides a corrected and unified list of seven transforms involving LHIs and Legendre functions, addressing normalization inconsistencies in existing tables.

Findings

Identified and corrected errors in standard integral tables.

Provided a merged, emended list of key transforms.

Discussed applications in communication theory.

Abstract

The complete Lipschitz-Hankel integrals (LHIs) include the Laplace transforms of the Bessel functions, multiplied by powers. Such Laplace transforms can be evaluated using associated Legendre functions. It is noted that there are errors in published versions of these evaluations, and a merged and emended list of seven transforms is given. Errata for standard reference works, such as the table of Gradshteyn and Ryzhik, are also given. Most of the errors are attributable to inconsistent normalization of the Legendre functions. These transforms can be viewed as limits of incomplete LHIs, which find application in communication theory.