{"paper":{"title":"The Evaluation of a Quartic Integral via Schwinger, Schur and Bessel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"And Christophe Vignat, Tewodros Amdeberhan, Victor H. Moll","submitted_at":"2010-09-13T14:02:03Z","abstract_excerpt":"We provide additional methods for the evaluation of the integral \\begin{eqnarray} N_{0,4}(a;m) & := & \\int_{0}^{\\infty} \\frac{dx} {\\left( x^{4} + 2ax^{2} + 1 \\right)^{m+1}} \\end{eqnarray} where $m \\in {\\mathbb{N}}$ and $a \\in (-1, \\infty)$ in the form \\begin{eqnarray} N_{0,4}(a;m) & = & \\frac{\\pi}{2^{m+3/2} (a+1)^{m+1/2} } P_{m}(a) \\end{eqnarray} where $P_{m}(a)$ is a polynomial in $a$. The first one is based on a method of Schwinger to evaluate integrals appearing in Feynman diagrams, the second one is a byproduct of an expression for a rational integral in terms of Schur functions. Finally, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2399","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"}