= Weighted identity for Howard's semicircle theorem
{title2=$\int(U-c)^2(|F'|^2+k^2|F|^2)\,dz=0$}
For a nonreal regular <Rayleigh equation for inviscid shear flow> mode vanishing at the walls, put $F=\widehat w/(U-c)$. The equation becomes $[(U-c)^2F']'-k^2(U-c)^2F=0$. Multiplication by $\overline F$ and <integration by parts> gives the displayed identity. With $Q=|F'|^2+k^2|F|^2$, its imaginary and real parts give $\int UQ=c_r\int Q$ and $\int U^2Q=|c|^2\int Q$. Since $(U-U_{\min})(U_{\max}-U)\ge0$, these imply <Howard's semicircle theorem>.
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