We present a calculation of the energy-density correlation ${d}_{\mathrm{\ensuremath{-}}}$(r,h) of the two-dimensional Ising model in a magnetic field under the limit T\ensuremath{\rightarrow}
Tc-, H\ensuremath{\rightarrow}0, and R\ensuremath{\rightarrow}\ensuremath{\infty} where h=const\ifmmode\times\else\texttimes\fi{}H\ensuremath{\Vert}T-
Tc${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}15/8}$ is fixed and small, while r=R\ensuremath{\Vert}T-
Tc\ensuremath{\Vert} is fixed and large. We find ${\mathit{d}}_{\mathrm{\ensuremath{-}}}$(r,h)\ensuremath{\sim}2${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{hr}}^{\mathrm{\ensuremath{-}}1/2}$${\mathrm{e}}^{\mathrm{\ensuremath{-}}2}$, where ${\ensuremath{\lambda}}_{l}$ are the positive solutions of
J1/3(${\ensuremath{\lambda}}_{\mathrm{l}}$/3)+${\mathrm{J}}_{\mathrm{\ensuremath{-}}1/3}$ (${\ensuremath{\lambda}}_{1/3}$)=0.
The study of the equation of state of the two-dimensional Ising model is initiated by use of the recently calculated
n-point spin-correlation functions. In the scaling region where $T\ensuremath{\rightarrow}{T}_{c}$ and $h=B{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{15}{8}}(\frac{H}{k{T}_{c}})$ is of order 1 (
B is a lattice-dependent constant) the free energy below
Tc is of the form $f(T, h)\ensuremath{-}f(T, 0)=\mathrm{const}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{2}$ $\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}(\frac{1}{n!}){C}_{n}^{\ensuremath{-}}{h}^{n}$ where the lattice-dependent constant is chosen so that ${C}_{1}^{\ensuremath{-}}=1$. The constant ${C}_{2}^{\ensuremath{-}}$ has been studied previously. In this paper we study ${C}_{3}^{\ensuremath{-}}$ and ${C}_{4}^{\ensuremath{-}}$ and find as a first approximation $\frac{{C}_{3}^{\ensuremath{-}}}{3!}\ensuremath{\sim}\ensuremath{-}\frac{11}{72{\ensuremath{\pi}}^{2}}$ and $\frac{{C}_{4}^{\ensuremath{-}}}{4!}\ensuremath{\sim}\frac{5}{189\ensuremath{\pi}}+\frac{59}{162{\ensuremath{\pi}}^{3}}$. These are in close agreement with the low-temperature-series values of -0.01538 and 0.0195.
Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.