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

Shingo Takeuchi

doi:10.1007/s11139-018-9993-y

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

Shingo Takeuchi

Research output: Contribution to journal › Article › peer-review

19 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 π_p. It is shown, only for p= 4 , that the generalized p-elliptic integrals yield a computation formula of π_p in terms of the arithmetic–geometric mean. This is a π_p-version of the celebrated formula of π, independently proved by Brent and Salamin in 1976.

Original language	English
Pages (from-to)	309-321
Number of pages	13
Journal	Ramanujan Journal
Volume	46
Issue number	2
DOIs	https://doi.org/10.1007/s11139-018-9993-y
Publication status	Published - 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

Access to Document

10.1007/s11139-018-9993-y

Cite this

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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.",

keywords = "Arithmetic–geometric mean, Brent–Salamin{\textquoteright}s algorithm, Complete elliptic integrals, Generalized trigonometric functions, p-Laplacian",

author = "Shingo Takeuchi",

note = "Funding Information: This work was supported by JSPS KAKENHI Grant Number 24540218. Publisher Copyright: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature.",

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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.

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KW - Brent–Salamin’s algorithm

KW - Complete elliptic integrals

KW - Generalized trigonometric functions

KW - p-Laplacian

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