Rayleigh equation for inviscid shear flow

For a parallel inviscid base flow $U(z)$ and disturbance streamfunction $\varphi (z)e^{i\alpha (x-ct)}$, the Rayleigh equation is

An inviscid parallel shear flow can possess an exponentially growing normal mode only if $U^{\prime \prime }$ changes sign somewhere in the flow.

If an inviscid shear-flow mode has complex wave speed $c$ and $U_{-}\le U\le U_{+}$, then