  Making use of the analytical properties of the $S$-matrix and a theorem of
Mittag-Leffler, model independent non-relativistic expressions for cross
sections in single channel elastic scattering, scattering phase shifts and
survival probabilities of resonances are derived. Provided certain conditions
are satisfied by the poles, the residues can also be estimated analytically.
Considerations of the low energy behaviour of the $S$-matrix and cross sections
reveal additional conditions on the residues of the poles appearing in the
Mittag-Leffler expansions. The exact expressions for the resonant cross section
and phase shift are shown to reduce to the commonly used Breit-Wigner formula
plus corrections. The latter is shown to approach the exact result with the
example of a meson and a baryon resonance. Finally, a comparison of the exact
expressions with some realistic examples is presented. The calculations of
survival probabilities in particular reveal the reason behind the
non-observability of non-exponential decay.
