Yeah, that part wasn't right. What I meant was Fourier expansions of periodic approximations, like taking this:

(a periodic approximation function for the given function)

which has imaginary period , then expand it either as a Fourier series along a line like , which is to the "right" of the singularity (or singularities when dealing with the approximations), or expand it along one like , which is to the "left", then continuum-sum one of those Fourier series and take the limit at infinite period. When the resulting functions are analytically continued by the continuum-sum recurrence equations to the whole plane, they should yield continuum sums with singularities going to the left and right, respectively.

(a periodic approximation function for the given function)

which has imaginary period , then expand it either as a Fourier series along a line like , which is to the "right" of the singularity (or singularities when dealing with the approximations), or expand it along one like , which is to the "left", then continuum-sum one of those Fourier series and take the limit at infinite period. When the resulting functions are analytically continued by the continuum-sum recurrence equations to the whole plane, they should yield continuum sums with singularities going to the left and right, respectively.