We revisit the problem of analytically computing the one point functions for
scalar fields in planar AdS black holes of arbitrary dimension, which are
sourced by the Weyl squared tensor. We analyze the problem in terms of power
series expansions around the boundary using the method of Frobenius. We clarify
the pole structure of the final answer in terms of operator mixing, as argued
previously by Grinberg and Maldacena. We generalize the techniques to also
obtain analytic results for slowly modulated spatially varying sources to first
non-trivial order in the wave vector for arbitrary dimension. We also study the
first order corrections to the one point function of the global AdS black hole
at large mass, where we perturb in terms that correspond to the curvature of
the horizon.
