A new form of the generalized complete elliptic integrals

Research output: Contribution to journalArticle

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Abstract

Generalized trigonometric functions are applied to Legendre’s form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can be easily shown that these integrals have similar properties to the classical ones. In particular, it is possible to establish a computation formula of the generalized p in terms of the arithmeticgeometric mean, in the classical way as the Gauss-Legendre algorithm for p by Brent and Salamin. Moreover, an elementary alternative proof of Ramanujan’s cubic transformation is also given.

LanguageEnglish
Pages202-226
Number of pages25
JournalKodai Mathematical Journal
Volume39
Issue number1
DOIs
StatePublished - 2016 Mar 25

Keywords

  • Arithmetic-geometric mean
  • Gauss-Legendre’s algorithm
  • Generalized complete elliptic integrals
  • Generalized trigonometric functions
  • P-Laplacian
  • Ramanujan’s cubic transformation

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A new form of the generalized complete elliptic integrals. / Takeuchi, Shingo.

In: Kodai Mathematical Journal, Vol. 39, No. 1, 25.03.2016, p. 202-226.

Research output: Contribution to journalArticle

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