  A Herglotz function is a holomorphic map from the open complex unit disk into
the closed complex right halfplane. A classical Herglotz function has an
integral representation against a positive measure on the unit circle. We prove
a free analytic analogue of the Herglotz representation and describe how our
representations specialize to the free probabilistic case. We also show that
the set of representable Herglotz functions arising from noncommutative
conditional expectations must be closed in a natural topology.
