A new form of the generalized complete elliptic integrals

DOI10.2996/KMJ/1458651700zbMATH Open1347.33040arXiv1411.4778OpenAlexW3105921886MaRDI QIDQ291010

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Publication date: 6 June 2016

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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 $pi$ in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for $pi$ by Salamin and Brent. Moreover, an elementary new proof of Ramanujan's cubic transformation is also given.

Full work available at URL: https://arxiv.org/abs/1411.4778

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