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

Research output: Contribution to journalArticle

5 Citations (Scopus)

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

Cite this

Complete p-elliptic integrals and a computation formula of πp for p= 4. / Takeuchi, Shingo.

In: Ramanujan Journal, Vol. 46, No. 2, 01.06.2018, p. 309-321.

Research output: Contribution to journalArticle

@article{1b8a1549fa98428d807318b39a777f94,
title = "Complete p-elliptic integrals and a computation formula of πp for p= 4",
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.",
keywords = "Arithmetic–geometric mean, Brent–Salamin’s algorithm, Complete elliptic integrals, Generalized trigonometric functions, p-Laplacian",
author = "Shingo Takeuchi",
year = "2018",
month = "6",
day = "1",
doi = "10.1007/s11139-018-9993-y",
language = "English",
volume = "46",
pages = "309--321",
journal = "Ramanujan Journal",
issn = "1382-4090",
publisher = "Springer Netherlands",
number = "2",

}

TY - JOUR

T1 - Complete p-elliptic integrals and a computation formula of πp for p= 4

AU - Takeuchi, Shingo

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

KW - Arithmetic–geometric mean

KW - Brent–Salamin’s algorithm

KW - Complete elliptic integrals

KW - Generalized trigonometric functions

KW - p-Laplacian

UR - http://www.scopus.com/inward/record.url?scp=85045150378&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045150378&partnerID=8YFLogxK

U2 - 10.1007/s11139-018-9993-y

DO - 10.1007/s11139-018-9993-y

M3 - Article

AN - SCOPUS:85045150378

VL - 46

SP - 309

EP - 321

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

IS - 2

ER -