Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

We compute exactly the spin-spin correlation functions 〈₀, ₀₌, ₍〉 for the two-dimensional Ising model on a square lattice in zero magnetic field for T>T₂ and T<T₂. We then analyze the correlation functions in the scaling limit TT₂, M^2+N^2 such that (T-T₂) is fixed. In this scaling limit 〈₀, ₀₌, ₍〉=R^-1{4}F_ (t) +R^-5{4}F₁ (t) +o (R^-5{4}), where t is the scaling variable R and F_ (t) and F₁ (t) are the scaling functions (is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-t behavior. A table of values for F_ (t) (good to ten significant digits) is also given. As an application we computer the coefficients C₀ and C₁ in the expansion k₁T (T) =C₀|1-{T₂T|}^-7{4}+C₁|1-{T₂T|}^-3{4}+O (1) of the zero-field susceptibility (T) as TT₂^.