  In \cite{GUW} we introduced a class of "semi-classical functions of isotropic
type", starting with a model case and applying Fourier integral operators
associated with canonical transformations. These functions are a substantial
generalization of the "oscillatory functions of Lagrangian type" that have
played major role in semi-classical and micro-local analysis. In this paper we
exhibit more clearly the nature of these isotropic functions by obtaining
oscillatory integral expressions for them. Then we use these to prove that the
classes of isotropic functions are equivariant with respect to the action of
general FIOs (under the usual clean-intersection hypothesis). The simplest
examples of isotropic states are the "coherent states", a class of oscillatory
functions that has played a pivotal role in mathematics and theoretical physics
beginning with their introduction by of Schr\"odinger in the 1920's. We prove
that every oscillatory function of isotropic type can be expressed as a
superposition of coherent states, and examine some implications of that fact.
We also show that certain functions of elliptic operators have isotropic
functions for Schwartz kernels. This lead us to a result on an eigenvalue
counting function that appears to be new (Corollary \ref{cor:altWeyl}).
