The complex Langevin method is a general method to treat systems with complex
action, such as QCD at nonzero density. The formal justification relies on the
absence of certain boundary terms, both at infinity and at the unavoidable poles
of the drift force. Here I focus on the boundary terms at these poles for simple
models, which so far have not been discussed in detail. The main result is that
those boundary terms (for the ``un-evolved'' observables) arise after running
the Langevin process for a finite time and vanish again as the Langevin time
goes to infinity. This is in contrast to the boundary terms at infinity, which
can be found to occur in the long time limit (cf.~the contribution by D\'enes
Sexty).