Factorization of multiple integrals representing the density matrix of a finite segment of the Heisenberg spin chain

We consider the inhomogeneous generalization of the density matrix of a finite segment of length m of the antiferromagnetic Heisenberg chain. It is a function of the temperature T and the external magnetic field h, and further depends on m 'spectral parameters' xi(j). For short segments of length 2 and 3 we decompose the known multiple integrals for the elements of the density matrix into finite sums over products of single integrals. This provides new numerically efficient expressions for the two-point functions of the in finite Heisenberg chain at short distances. It further leads us to conjecture an exponential formula for the density matrix involving only a double Cauchy-type integral in the exponent. We expect this formula to hold for arbitrary m and T but zero magnetic field.