The bateman functions revisited after 90 years—a survey of old and new results

Alexander Apelblat, Armando Consiglio, Francesco Mainardi

Research output: Contribution to journalReview articlepeer-review

1 Scopus citations


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.

Original languageEnglish
Article number1273
Issue number11
StatePublished - 1 Jun 2021


  • Bateman functions
  • Confluent hypergeo-metric functions
  • Havelock functions
  • Integral-bateman functions

ASJC Scopus subject areas

  • General Mathematics


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