Equivalence of the two results for the free energy of the chiral Potts model

The free energy of the chiral Potts model has been obtained in two ways. The first used only the star-triangle relation, symmetries and invariances, and led to a system of equations that implicitly define the free energy, and from which the critical behaviour can be obtained. The second used the functional relations derived by Bazhanov and Stroganov, solving them to obtain the free energy explicitly as a double integral. Here we obtain, for the first time, a direct verification that the two results are identical at all temperatures.