{"paper":{"title":"The complete $p$-elliptic integrals and a computation formula of $\\pi_p$ for $p=4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.NT"],"primary_cat":"math.CA","authors_text":"Shingo Takeuchi","submitted_at":"2015-03-09T08:38:58Z","abstract_excerpt":"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 Salamin and Brent in 1976."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02394","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}