text: The Fano resonance line-shape is due to interference between two scattering amplitudes, one due to scattering within a continuum of states (the background process) and the second due to an excitation of a discrete state (the resonant process). The energy of the resonant state must lie in the energy range of the continuum (background) states for the effect to occur. Near the resonant energy, the background scattering amplitude typically varies slowly with energy while the resonant scattering amplitude changes both in magnitude and phase quickly. It is this variation that creates the asymmetric profile.
  For energies far from the resonant energy  the background scattering process dominates. Within  of the resonant energy, the phase of the resonant scattering amplitude changes by . It is this rapid variation in phase that creates the asymmetric line-shape.
  Fano showed that the total scattering cross-section  assumes the following form,
  where  describes the line width of the resonant energy and , the Fano parameter, measures the ratio of resonant scattering to the direct (background) scattering amplitude. This is consistent with the interpretation within the Feshbach–Fano partitioning theory. In the case the direct scattering amplitude vanishes, the  parameter becomes zero and the Fano formula becomes :
  Looking at transmission shows that this last expression boils down to the expected Breit–Wigner (Lorentzian) formula, as , the three parameters Lorentzian function (note that it is not a density function and does not integrate to 1, as its amplitude  is 1 and not ).