  The analytic properties in the energy variable k_{0} of finite-temperature
self-energies are investigated. A typical branch cut results from n particles
being emitted into the heat bath and n' being absorbed from the heat bath.
There are three main results: First, in addition to the branch points at which
the cuts terminate, there are also branch points attached to the cuts along
their length. Second, branch points at k_{0}=\pm k are ubiquitous and for
massive particles they are essential singularities. Third, in a perturbative
expansion using free particle propagators or in a resummed expansion in which
the propagator pole occurs at a real energy, the self-energy will have a branch
point at the pole location.
