Painlevé functions of the third kind

We explicitly construct the one-parameter family of solutions, η (ϑ;ν,λ), that remain bounded as ϑ→∞ along the positive real ϑ axis for the Painlevé equation of third kind ww′′= (w′)2−ϑ−1ww′+2νϑ−1(w3−w) +w4−1, where, as ϑ→∞, η ∼ 1−λΓ (ν+1/2)2−2νϑ−ν−1/2e−2ϑ. We further construct a representation for ψ (t;ν,λ) =−lnη (t/2;ν,λ), where ψ (t;ν,λ) satisfies the differential equation ψ′′+t−1ψ′= (1/2)sinh(2ψ)+2νt−1 sinh(ψ). The small-ϑ behavior of η (ϑ;ν,λ) is described for ‖λ‖π−1 by η (ϑ;ν,λ) ∼ 2σBϑσ. The parameters σ and B are given as explicit functions of λ and ν. Finally an identity involving the Painlevé transcendent η (ϑ;ν,λ) is proved. These results for the special case ν=0 and λ=π−1 make rigorous the analysis of the scaling limit of the spin–spin correlation function of the two-dimensional Ising model.