  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 infinite 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.
