Two-matrix model with semiclassical potentials and extended Whitham hierarchy

We consider the two-matrix model with potentials whose derivatives are arbitrary rational functions of fixed pole structure and the support of the spectra of the matrices are union of intervals (hard edges). We derive an explicit formula for the planar limit of the free energy and we derive a calculus which allows us to compute derivatives of arbitrarily high order by extending classical Rauch's variational formulae. The four-point correlation functions are explicitly worked out. The formalism extends naturally to the computation of residue formulae for the tau function of the so-called universal Whitham hierarchy studied mainly by I Krichever: our setting extends the moduli space in that there are certain extra data.