= Rayleigh stability integral identity
{c}
{title2=$\omega^2=k^2\int\kappa_{\rm ep}^2|u|^2dr/\int(|u'|^2+k^2|u|^2)dr$}
Multiply the <narrow-gap centrifugal stability equation> by the complex conjugate mode and use <integration by parts> with zero endpoint radial velocities. This gives the displayed real <Rayleigh quotient>. Its denominator is positive. Positive <Rayleigh discriminant> throughout the gap gives positive $\omega^2$ for every nonzero mode, excluding exponential axisymmetric growth; nonnegativity permits marginal cases. In the alternative convention $\Phi=-\kappa_{\rm ep}^2$, the numerator is $-k^2\int\Phi|u|^2dr$, so the stable sign is $\Phi<0$. For rigid rotation, $\kappa_{\rm ep}^2=4\Omega^2$ gives $\omega_n^2=4\Omega^2k^2/[k^2+(n\pi/d)^2]$.
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