Painlevé functions of the third kind

Abstract

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) =−ln[η (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) ∼ 2^σBϑ^σ. 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.