Asymptotic expansion for the resistance between two maximally separated nodes on an M by N resistor network

We analyze the exact formulas for the resistance between two arbitrary notes in a rectangular network of resistors under free, periodic and cylindrical boundary conditions obtained by Wu [J. Phys. A 37, 6653 (2004)]. Based on such expression, we then apply the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of the resistance between two maximally separated nodes on an M×N rectangular network of resistors with resistors r and s in the two spatial directions. Our results is $1/s{R}_{M×{}N}(r,s)=c({ρ})ln S+{c}₀({ρ},{ξ})+{{∑}}ₚ₌₁^{{∞}}{{c}₂ₚ({ρ},{ξ})}{{S}ᵖ}$ with $S=MN$, ${ρ}=r/s$ and ${ξ}=M/N$. The all coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ${{ξ}}eff=√{{ρ}}{ξ}$ for free and periodic boundary conditions and ${{ξ}}eff=√{{ρ}}{ξ}/2$ for cylindrical boundary condition and show that all finite-size correction terms are invariant under transformation ${{ξ}}eff{→}1/{{ξ}}eff$.