### Abstract

The complete p-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function sin _{p}θ 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 | |

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

### Cite this

**Complete p-elliptic integrals and a computation formula of π _{p} for p= 4.** / Takeuchi, Shingo.

Research output: Contribution to journal › Article

_{p}for p= 4',

*Ramanujan Journal*, vol. 46, no. 2, pp. 309-321. https://doi.org/10.1007/s11139-018-9993-y

}

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

VL - 46

SP - 309

EP - 321

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

IS - 2

ER -