The Ramanujan Journal

, Volume 46, Issue 2, pp 309–321 | Cite as

Complete p-elliptic integrals and a computation formula of \(\pi _p\) for \(p=4\)

  • Shingo Takeuchi


The complete p-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function \(\sin _p{\theta }\) and its half-period \(\pi _p\). It is shown, only for \(p=4\), that the generalized p-elliptic integrals yield a computation formula of \(\pi _p\) in terms of the arithmetic–geometric mean. This is a \(\pi _p\)-version of the celebrated formula of \(\pi \), independently proved by Brent and Salamin in 1976.


Generalized trigonometric functions Complete elliptic integrals Arithmetic–geometric mean Brent–Salamin’s algorithm p-Laplacian 

Mathematics Subject Classification

33E05 33C75 11Z05 



The author would like to thank the anonymous reviewers for his/her valuable comments and suggestions to improve the quality of the paper.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesShibaura Institute of TechnologySaitamaJapan

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