  The trace of integer powers of the local density matrix corresponding to the
vacuum state reduced to a region V can be formally expressed in terms of a
functional integral on a manifold with conical singularities. Recently, some
progress has been made in explicitly evaluating this type of integrals for free
fields. However, finding the associated geometric entropy remained in general a
difficult task involving an analytic continuation in the conical angle. In this
paper, we obtain this analytic continuation explicitly exploiting a relation
between the functional integral formulas and the Chung-Peschel expressions for
the density matrix in terms of correlators. The result is that the entropy is
given in terms of a functional integral in flat Euclidean space with a cut on V
where a specific boundary condition is imposed. As an example we get the exact
entanglement entropies for massive scalar and Dirac free fields in 1+1
dimensions in terms of the solutions of a non linear differential equation of
the Painleve V type.
