dorsal/arxiv
View SchemaOn the Solution of a Painlev\'e III Equation
| Authors | Harold Widom |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9808015 |
| URL | https://arxiv.org/abs/solv-int/9808015 |
Abstract
In a 1977 paper of McCoy, Tracy and Wu there appeared for the first time the solution of a Painlev\'e equation in terms of Fredholm determinants of integral operators. This equation is $\psi''(t)+t^{-1}\psi'(t)=(1/2) \sinh 2\psi+2\alpha t^{-1} \sinh\psi$, a special case of the Painlev\'e III equation. The proof in the cited paper is complicated, and the purpose of this note is to give a more straightforward one. First we give an equivalent formulation of the solution in terms of the kernel ${e^{-t (x+x^{-1})/2}\over x+y}\Big|{x-1\over x+1}\Big|^{2\alpha}$. There are already in the literature relatively simple proofs of the fact that when $\alpha=0$ Fredholm determinants of this kernel give solutions to the equation. We extend this result here to general $\alpha$.
{
"annotation_id": "a7e8991e-e65e-41fa-948f-3e2e97bede7c",
"date_created": "2026-03-02T18:02:51.459000Z",
"date_modified": "2026-03-02T18:02:51.459000Z",
"file_hash": "70deb958d5bb297422a1c98292afb06c29aea01ac1fcb3c866eb9fb5d64b9449",
"private": false,
"record": {
"abstract": "In a 1977 paper of McCoy, Tracy and Wu there appeared for the first time the\nsolution of a Painlev\\\u0027e equation in terms of Fredholm determinants of integral\noperators. This equation is $\\psi\u0027\u0027(t)+t^{-1}\\psi\u0027(t)=(1/2) \\sinh 2\\psi+2\\alpha\nt^{-1} \\sinh\\psi$, a special case of the Painlev\\\u0027e III equation. The proof in\nthe cited paper is complicated, and the purpose of this note is to give a more\nstraightforward one. First we give an equivalent formulation of the solution in\nterms of the kernel ${e^{-t (x+x^{-1})/2}\\over x+y}\\Big|{x-1\\over\nx+1}\\Big|^{2\\alpha}$. There are already in the literature relatively simple\nproofs of the fact that when $\\alpha=0$ Fredholm determinants of this kernel\ngive solutions to the equation. We extend this result here to general $\\alpha$.",
"arxiv_id": "solv-int/9808015",
"authors": [
"Harold Widom"
],
"categories": [
"solv-int",
"math.FA",
"nlin.SI"
],
"title": "On the Solution of a Painlev\\\u0027e III Equation",
"url": "https://arxiv.org/abs/solv-int/9808015"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "b96c0235-015c-4956-8ae6-446e5d4fde2d",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}