A new form of the generalized complete elliptic integrals
Classical Analysis and ODEs
2019-03-12 v3 Analysis of PDEs
Number Theory
Abstract
Generalized trigonometric functions are applied to the Legendre-Jacobi standard 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 in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for by Salamin and Brent. Moreover, an elementary new proof of Ramanujan's cubic transformation is also given.
Cite
@article{arxiv.1411.4778,
title = {A new form of the generalized complete elliptic integrals},
author = {Shingo Takeuchi},
journal= {arXiv preprint arXiv:1411.4778},
year = {2019}
}
Comments
29 pages