{"paper":{"title":"Rational Solutions of the Painlev\\'e-III Equation: Large Parameter Asymptotics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP","nlin.SI"],"primary_cat":"math.CA","authors_text":"Peter D. Miller, Thomas Bothner","submitted_at":"2018-08-04T04:15:30Z","abstract_excerpt":"The Painlev\\'e-III equation with parameters $\\Theta_0=n+m$ and $\\Theta_\\infty=m-n+1$ has a unique rational solution $u(x)=u_n(x;m)$ with $u_n(\\infty;m)=1$ whenever $n\\in\\mathbb{Z}$. Using a Riemann-Hilbert representation proposed in \\cite{BothnerMS18}, we study the asymptotic behavior of $u_n(x;m)$ in the limit $n\\to+\\infty$ with $m\\in\\mathbb{C}$ held fixed. We isolate an eye-shaped domain $E$ in the $y=n^{-1}x$ plane that asymptotically confines the poles and zeros of $u_n(x;m)$ for all values of the second parameter $m$. We then show that unless $m$ is a half-integer, the interior of $E$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.01421","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"}