The Bateman Functions Revisited After 90 Years -- A Survey of Old and New Results

TL;DR

This paper revisits the Bateman and Havelock functions, providing a comprehensive survey of their properties, recent results, and potential generalizations, aiming to revive interest in these historically significant special functions.

Contribution

It offers a detailed review of classical properties and introduces new results for Bateman and Havelock functions, including their generalizations and related integral functions.

Findings

Compiled fundamental properties and relations of Bateman functions.

Presented new results and generalizations of these functions.

Highlighted the significance of integer order cases and related functions.

Abstract

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to known in the literature generalizations of these functions and to the Bateman-integral function.