{"paper":{"title":"Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"D.Karp, S.M.Sitnik","submitted_at":"2006-04-03T09:57:45Z","abstract_excerpt":"We find two convergent series expansions for Legendre's first incomplete elliptic integral $F(\\lambda,k)$ in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square $0<\\lambda,k<1$. Truncated expansions yield asymptotic approximations for $F(\\lambda,k)$ as $\\lambda$ and/or $k$ tend to unity, including the case when logarithmic singularity $\\lambda=k=1$ is approached from any direction. Explicit error bounds are given at every order of approximation. For the reader's convenience we present explicit expressions for low-order approximations "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0604026","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"}