Yeah, that part wasn't right. What I meant was Fourier expansions of periodic approximations, like taking this:
\( f_u(z) = \frac{1}{1 + \left(u \sinh\left(\frac{z}{u}\right)\right)},\ u \in \mathbb{R} \)
(a periodic approximation function for the given function)
which has imaginary period \( 2\pi i u \), then expand it either as a Fourier series along a line like \( \Re(z) = 0 \), which is to the "right" of the singularity (or singularities when dealing with the approximations), or expand it along one like \( \Re(z) = -2 \), 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.
\( f_u(z) = \frac{1}{1 + \left(u \sinh\left(\frac{z}{u}\right)\right)},\ u \in \mathbb{R} \)
(a periodic approximation function for the given function)
which has imaginary period \( 2\pi i u \), then expand it either as a Fourier series along a line like \( \Re(z) = 0 \), which is to the "right" of the singularity (or singularities when dealing with the approximations), or expand it along one like \( \Re(z) = -2 \), 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.