{"paper":{"title":"Some asymptotics for the Bessel functions with an explicit error term","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ilia Krasikov","submitted_at":"2011-07-11T12:09:57Z","abstract_excerpt":"We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds.\n  We will work out the details for the Bessel function $J_\\nu (x)$ and the Airy function $Ai(x)$ and find a sharp approximation for their zeros.\n  We also answer the question raised by Olenko by showing that\n  $$c_1 | \\nu^2-1/4\\,| < \\sup_{x \\ge 0} x^{3/2}|J_\\nu(x)-\\sqrt{\\frac{2}{\\pi x}} \\, \\cos (x-\\frac{\\pi \\nu}{2}-\\frac{\\pi}{4}\\,)| <c_2 |\\nu^2-1/4\\,|, $$ $ \\nu \\ge -1/2 \\, ,$ for some explicit nume"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2007","kind":"arxiv","version":2},"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"}