A new form of the generalized complete elliptic integrals

Kodai Mathematical Journal

Shingo Takeuchi¹ 

[1] Department of Mathematical Sciences Shibaura Institute of Technology

关键词: Generalized trigonometric functions;    Generalized complete elliptic integrals;    Ramanujan's cubic transformation;    Arithmetic-geometric mean;    Gauss-Legendre's algorithm;    p-Laplacian;   

DOI  :  10.2996/kmj/1458651700

学科分类：数学（综合）

来源: Tokyo Institute of Technology, Department of Mathematics

【 摘 要 】

References(33)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 π in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for π by Brent and Salamin. Moreover, an elementary alternative proof of Ramanujan's cubic transformation is also given.

【 授权许可】

Unknown