A formalism for working within the finite set of Matsubara frequencies that correspond to a regular discretization in imaginary time of a thermal Green’s function. A mathematically consistent scheme for such GF that are periodic in complex frequency space is developed, including a generalized Dyson equation and Luttinger-Ward functional.  The method allows for doing very accurate analytic continuation from imaginary to real frequencies. Paper. Swe_Swiss_110826.pdf