  We study the mechanisms setting the radius of convergence of hydrodynamic
dispersion relations in kinetic theory in the relaxation time approximation.
This introduces a qualitatively new feature with respect to holography: a
nonhydrodynamic sector represented by a branch cut in the retarded Green's
function. In contrast with existing holographic examples, we find that the
radius of convergence in the shear channel is set by a collision of the
hydrodynamic pole with a branch point. In the sound channel it is set by a
pole-pole collision on a non-principal sheet of the Green's function. More
generally, we examine the consequences of the implicit function theorem in
hydrodynamics and give a prescription to determine a set of points that
necessarily includes all complex singularities of the dispersion relation. This
may be used as a practical tool to assist in determining the radius of
convergence of hydrodynamic dispersion relations.
