Exact amplitude ratio and finite-size corrections for theM×Nsquare lattice Ising model

Let $f,\hspace{0.5em}U,$ and C represent, respectively, the free energy, the internal energy, and the specific heat of the critical Ising model on the $M\times N$ square lattice with periodic boundary conditions, and $f_{\infty }$ represents f for fixed $M/N$ and $\overrightarrow{N}\infty .$ We find that $f,\hspace{0.5em}U,$ and C can be written as $N\left(f-f_{\infty }\right)={\sum }_{i=1\mathrm{}}^{\infty }{f}_{2i-1}{/N}^{2i-1},\hspace{0.5em}U=-\sqrt{2}+{\sum }_{i=1\mathrm{}}^{\infty }{u}_{2i-1}{/N}^{2i-1},$ and $C=8\mathrm{}\ln N/\pi +{\sum }_{i=0\mathrm{}}^{\infty }{c}_i{/N}^i,$ i.e., $\mathrm{Nf}$ and U are odd functions of $N^{-1}.$ We also find that $u_{2i-1}{/c}_{2i-1}=1/\sqrt{2}$ and $u_{2i}{/c}_{2i}=0\mathrm{}$ for $1<~i<\mathrm{}\infty$ and obtain closed form expressions for $f,\hspace{0.5em}U,$ and C up to orders ${1/N}^5,\hspace{0.5em}{1/N}^5,$ and ${1/N}^3,$ respectively, which implies an analytic equation for $c_5.$