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.
| Original language | English |
|---|---|
| Pages (from-to) | 202-226 |
| Number of pages | 25 |
| Journal | Kodai Mathematical Journal |
| Volume | 39 |
| Issue number | 1 |
| DOIs | |
| Publication status | 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
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 journal › Article
}
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
JF - Kodai Mathematical Journal
SN - 0386-5991
IS - 1
ER -