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We analyze the exact formulae for the resistance between two arbitrary notesin a rectangular network of resistors under free, periodic and cylindricalboundary conditions obtained by Wu [J. Phys. A 37, 6653 (2004)]. Based on suchexpression, we then apply the algorithm of Ivashkevich, Izmailian and Hu [J.Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of theresistance between two maximum separated nodes on an $M \times N$ rectangularnetwork of resistors with resistors $r$ and $s$ in the two spatial directions.Our results is $ \frac{1}{s}R_{M\times N}(r,s)= c(\rho)\,\ln{S}+c_0(\rho,\xi)+\sum_{p=1}^{\infty} \frac{c_{2p}(\rho,\xi)}{S^{p}} $ with$S=M N$, $\rho=r/s$ and $\xi=M/N$. The all coefficients in this expansion areexpressed through analytical functions. We have introduced the effective aspectratio $\xi_{eff} = \sqrt{\rho}\;\xi$ for free and periodic boundary conditionsand $\xi_{eff} = \sqrt{\rho}\;\xi/2$ for cylindrical boundary condition andshow that all finite size correction terms are invariant under transformation$\xi_{eff} \to {1}/\xi_{eff}$.

Asymptotic expansion for the resistance between two maximum separated nodes on a $M \times N$ resistor network