A generalization of Poisson’s summation formula is derived – in a non-rigorous way – allowing evaluation of sums from 1 (or any finite integer) ¥ instead of the usual range -¥+¥. This is achieved in two ways, either by introducing a converging factor in a geometric series of exponential functions and letting it approach zero in a controlled way or by applying a Hilbert transform to the series. Several examples illustrate its usefulness in the evaluation of series and specific applications.