According to Onsager, the field-free partition function of a rectangular lattice (with interaction parameters \(J\) and \(J^{\prime}\) in the two perpendicular directions) is given by

\[
\frac{1}{N} \ln Q(T)=\ln 2+\frac{1}{2 \pi^{2}} \int_{0}^{\pi} \int_{0}^{\pi} \ln \left\{\cosh (2 \gamma) \cosh \left(2 \gamma^{\prime}\right)-\sinh (2 \gamma) \cos \omega-\sinh \left(2 \gamma^{\prime}\right) \cos \omega^{\prime}\right\} d \omega d \omega^{\prime}
\]

where \(\gamma=J / k T\) and \(\gamma^{\prime}=J^{\prime} / k T\). Show that if \(J^{\prime}=0\), this leads to expression (13.2.9) for the linear chain with \(B=0\) while if \(J^{\prime}=J\), one is led to expression (13.4.22) for the square net. Locate the critical point of the rectangular lattice and study its thermodynamic behavior in the neighborhood of that point.