1998/08/24 by Harold Widom, Widom, Harold
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #math.FA #nlin.SI #solv-int
paper · pdf · doi:10.48550/arxiv.solv-int/9808015
LaTeX file. 9 pages
arxiv created 1998/08/24 · arxiv updated 2009/11/30
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-1ψ'(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|x-1\over 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.