Using equations (13.4.22) and (13.4.28) at \(T=T_{c}\), show that the entropy of the two-dimensional Ising model on a square lattice at its critical point is given by

\[
\frac{S_{c}}{N k}=\frac{2 G}{\pi}+\frac{1}{2} \ln 2-\sqrt{ } 2 K_{c} \simeq 0.3065
\]

here, \(G\) is Catalan's constant, which is approximately equal to 0.915966 . Compare this result with the ones following from the Bragg-Williams approximation and from the Bethe approximation; see Problem 12.15.