The $XX$ and Ising Limits in Integral Formulas for Finite-Temperature Correlation Functions of the $XXZ$ Chain

We consider a multiple-integral representation for a one-parameter generating function of the finite temperature $S^z$–$S^z$ correlation functions of the antiferromagnetic spin-$1/2$ $XXZ$ chain in the $XX$ limit and in the Ising limit. We show how the multiple integrals reduce to single integrals in these limits, thus reproducing known results.