On the Solution of a Painlevé III Equation
Abstract
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 $α$.
LaTeX file. 9 pages
LaTeX file. 9 pages