Equivalence of the Falicov-Kimball and Brandt-Mielsch forms for the free   energy of the infinite-dimensional Falicov-Kimball model
  Falicov and Kimball proposed a real-axis form for the free energy of the Falicov-Kimball model that was modified for the coherent potential approximation by Plischke. Brandt and Mielsch proposed an imaginary-axis form for the free energy of the dynamical mean field theory solution of the Falicov-Kimball model. It has long been known that these two formulae are numerically equal to each other; an explicit derivation showing this equivalence is presented here. 
A. M. Shvaika (Institute for Condensed Matter Physics, Lviv, Ukraine)   and J. K. Freericks (Georgetown University)