On the Solution of a Painlevé III Equation
| dc.creator | Widom, Harold | |
| dc.date | 1998-08-24 | |
| dc.date.accessioned | 2026-07-07T06:17:39Z | |
| dc.date.available | 2026-07-07T06:17:39Z | |
| dc.description | In a 1977 paper of McCoy, Tracy and Wu there appeared for the first time the solution of a Painlevé equation in terms of Fredholm determinants of integral operators. This equation is $ψ''(t)+t^{-1}ψ'(t)=(1/2) \sinh 2ψ+2αt^{-1} \sinhψ$, a special case of the Painlevé 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 ${e^{-t (x+x^{-1})/2}\over x+y}\Big|{x-1\over x+1}\Big|^{2α}$. There are already in the literature relatively simple proofs of the fact that when $α=0$ Fredholm determinants of this kernel give solutions to the equation. We extend this result here to general $α$. | |
| dc.description | LaTeX file. 9 pages | |
| dc.identifier | https://arxiv.org/abs/solv-int/9808015 | |
| dc.identifier | http://arxiv.org/abs/solv-int/9808015 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/94454 | |
| dc.subject | Exactly Solvable and Integrable Systems | |
| dc.subject | Functional Analysis | |
| dc.title | On the Solution of a Painlevé III Equation | |
| dc.type | text |