Analytic properties of scattering amplitudes are used to develop Breit–Wigner parametrizations of resonant tunneling currents in heterostructures. The analysis reveals a problem in using conventional Breit–Wigner resonance expressions to describe the important regions of peak current and negative differential resistance which occur when resonances are near the emitter threshold. Commonly used Breit–Wigner expressions (a) do not incorporate the correct threshold branch points, (b) do not possess the correct limit at threshold, and (c) do not satisfy unitarity both above and below threshold. Expressions which avoid such difficulties are deduced and used to parametrize tunneling current formulas.