The random cluster model and a new integration identity

We evaluate the free energy of the random cluster model at its critical point for 0 < q < 4 using an exact result due to Baxter, Temperley and Ashley, and obtain an explicit expression in the form of an infinite series. It is found that the resulting series expression assumes a form which depends on whether or not π/2 cos-1(√q/2) is a rational number. As a by-product, our consideration leads to a closed-form evaluation of the integral 1/(4π2) ∫02πdΘ ∫02πdΦ ln[A+B+C - AcosΘ - BcosΦ - Ccos(Θ+Φ)] = -ln(2S) + (2/π)[Ti2(AS) + Ti2(BS) + Ti2(CS)], which arises in lattice statistics, where A, B, C ≥ 0 and S=1/√(AB + BC + CA)