On the Solution of a Painlev\'e III Equation

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 $\alpha$.