### 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.

Language | English |
---|---|

Pages | 202-226 |

Number of pages | 25 |

Journal | Kodai Mathematical Journal |

Volume | 39 |

Issue number | 1 |

DOIs | |

State | Published - 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

*Kodai Mathematical Journal*,

*39*(1), 202-226. DOI: 10.2996/kmj/1458651700

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

Research output: Research - peer-review › Article

*Kodai Mathematical Journal*, vol 39, no. 1, pp. 202-226. DOI: 10.2996/kmj/1458651700

}

TY - JOUR

T1 - A new form of the generalized complete elliptic integrals

AU - Takeuchi,Shingo

PY - 2016/3/25

Y1 - 2016/3/25

N2 - 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.

AB - 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.

KW - Arithmetic-geometric mean

KW - Gauss-Legendre’s algorithm

KW - Generalized complete elliptic integrals

KW - Generalized trigonometric functions

KW - P-Laplacian

KW - Ramanujan’s cubic transformation

UR - http://www.scopus.com/inward/record.url?scp=84961575124&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84961575124&partnerID=8YFLogxK

U2 - 10.2996/kmj/1458651700

DO - 10.2996/kmj/1458651700

M3 - Article

VL - 39

SP - 202

EP - 226

JO - Kodai Mathematical Journal

T2 - Kodai Mathematical Journal

JF - Kodai Mathematical Journal

SN - 0386-5991

IS - 1

ER -