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The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results

Author

Listed:
  • Alexander Apelblat

    (Department of Chemical Engineering, Ben Gurion University of the Negev, Beer Sheva 84105, Israel)

  • Armando Consiglio

    (Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, D-97074 Würzburg, Germany)

  • Francesco Mainardi

    (Dipartimento di Fisica e Astronomia, Università di Bologna, & INFN, Via Irnerio 46, I-40126 Bologna, Italy)

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 generalizations of these functions and to the Bateman-integral function known in the literature.

Suggested Citation

  • Alexander Apelblat & Armando Consiglio & Francesco Mainardi, 2021. "The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results," Mathematics, MDPI, vol. 9(11), pages 1-27, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1273-:d:567001
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