  In this paper, we give subordination functions for free additive and free
multiplicative deconvolutions in some domain of the complex half-plane, under
the condition that the distributions admit moments, respectively, of second
order for the additive deconvolution and of fourth order for the multiplicative
one. Our method of proof allows us to give an algorithm to calculate these
subordinations functions, and thus the associated Cauchy transforms, for
complex numbers with imaginary part bigger than a parameter depending on the
measure to deconvolve. This reduces the problem of free deconvolutions to the
one of the classical deconvolution with a Cauchy distribution and thus combined
with known methods for the latter problem we are able to solve the
deconvolution problem for the scalar case. We present also an extension of
these results to the case of operator valued distributions.
