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
ψ
′′
(t)+
t
−1
ψ
′
(t)=(1/2)sinh2ψ+2α
t
−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
x+y
∣
∣
x−1
x+1
∣
∣
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
.