= Solution
Choose $a=1/2$ in the <power-transformed Taylor–Goldstein energy identity> and let $Q=N^2-(U')^2/4$. The result is
$$
\int V(|q'|^2+k^2|q|^2)dz=\int\left(\frac QV-\frac{U''}2\right)|q|^2dz.
$$
Because $\operatorname{Im}V=-c_i$ and $\operatorname{Im}(1/V)=c_i/|V|^2$, its <imaginary part> gives
$$
c_i\int_{-L}^L\left[|q'|^2+k^2|q|^2+\frac{Q}{|U-c|^2}|q|^2\right]dz=0.
$$
If $4N^2>(U')^2$ everywhere, the integral is strictly positive for every nonzero <eigenfunction>, so a mode with $c_i\ne0$ is impossible. Thus the flow is neutrally stable to these inviscid <normal modes>. This is the <Miles–Howard theorem>, expressed as a lower bound $\mathrm{Ri}>1/4$ on the <gradient Richardson number>. It excludes exponential modal growth; it does not by itself exclude <transient growth>.
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