I have already asked this at MSE but did not get an answer.

In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. Mathematically, these derivations are somewhat magical (typically one inserts an "infinitesimally small" iε term, and then the different propagators result from different integration contours around certain poles). On the other hand, there is a mathematically rigorous theory of fundamental solutions to PDEs, but I have never seen anything analogous to these propagators in a PDE book. Can somebody recommend a source (book/lecture notes/paper), where the retarded, advanced and Feynman propagators are treated in a mathematically rigorous way, so that I can see the connection between QFT and PDE theory?