{"paper":{"title":"A note on an integral associated with the Kelvin ship-wave pattern","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R. B. Paris","submitted_at":"2015-07-08T15:19:03Z","abstract_excerpt":"The velocity potential in the Kelvin ship-wave source can be partly expressed in terms of space derivatives of the single integral \\[F(x,\\rho,\\alpha)=\\int_{-\\infty}^\\infty \\exp\\,[-\\frac{1}{2}\\rho \\cosh (2u-i\\alpha)] \\cos (x\\cosh u)\\,du,\\] where $(x, \\rho, \\alpha)$ are cylindrical polar coordinates with origin based at the source and $-\\pi/2\\leq\\alpha\\leq\\pi/2$. An asymptotic expansion of $F(x,\\rho,\\alpha)$ when $x$ and $\\rho$ are small, but such that $M\\equiv x^2/(4\\rho)$ is large, was given using a non-rigorous approach by Bessho in 1964 as a sum involving products of Bessel functions. This e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02193","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"}