On the Solution of a Painlevé III Equation

Abstract

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 ${\psi }^{\prime \prime }(t)+t^{-1}{\psi }^{\prime }(t)=(1/2)\sinh 2\psi +2\alpha t^{-1}\sinh \psi$, 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 $\frac{e^{-t(x+x^{-1})/2}}{x+y}|\frac{x-1}{x+1}{|}^{2\alpha }$. 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 $\alpha$.

LaTeX file. 9 pages