High-precision computation of the confluent hypergeometric functions via Franklin-Friedman expansion

Abstract

We present a method of high-precision computation of the confluent hypergeometric functions using an effective computational approach of what we termed Franklin-Friedman expansions. These expansions are convergent under mild conditions of the involved amplitude function and for some interesting cases the coefficients can be rapidly computed, thus providing a viable alternative to the conventional dichotomy between series expansion and asymptotic expansion. The present method has been extensively tested in different regimes of the parameters and compared with recently investigated convergent and uniform asymptotic expansions.

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Acknowledgments

The author acknowledges support from Ministerio de Economía, Industria y Competitividad, project APCOM (TIN2014-57226-P)

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Correspondence to Guillermo Navas-Palencia.

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Communicated by: Raymond H. Chan

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Navas-Palencia, G. High-precision computation of the confluent hypergeometric functions via Franklin-Friedman expansion. Adv Comput Math 44, 841–859 (2018). https://doi.org/10.1007/s10444-017-9565-5

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Keywords

  • Confluent hypergeometric functions
  • Franklin-friedman expansion
  • Uniform series expansion
  • Arbitrary-precision arithmetic

Mathematics Subject Classifications (2010)

  • 33C15
  • 33F05
  • 41A58
  • 65D20
  • 68W30