Frequency Cusp, a Means for Discriminating between Convective and Nonconvective Instability

The colliding-pole criterion for determining instability types from dispersion relations calls for a process of conformal mapping from the frequency plane into the wave-number plane. It is shown that this process can be inverted by mapping lines, parallel to the imaginary axis, from the wave-number plane into the frequency plane. Lines mapping into a cusp locate the frequencies at which the group velocity vanishes. These frequencies determine the instability character of the system.