Complete p-elliptic integrals and a computation formula of πp for p= 4

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Abstract

The complete p-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function sin pθ and its half-period πp. It is shown, only for p= 4 , that the generalized p-elliptic integrals yield a computation formula of πp in terms of the arithmetic–geometric mean. This is a πp-version of the celebrated formula of π, independently proved by Brent and Salamin in 1976.

Original languageEnglish
Pages (from-to)309-321
Number of pages13
JournalRamanujan Journal
Volume46
Issue number2
DOIs
Publication statusPublished - 2018 Jun 1

Keywords

  • Arithmetic–geometric mean
  • Brent–Salamin’s algorithm
  • Complete elliptic integrals
  • Generalized trigonometric functions
  • p-Laplacian

ASJC Scopus subject areas

  • Algebra and Number Theory

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