Positivity condition for a phase-locked standing wave ($z=|A{|}^2>0$)

Writing $U=\mathrm{Re}(\beta +\gamma )$ and $V=\mathrm{Im}(\beta +\gamma )$, the squared amplitude is $z_{\pm }=[\mu U\pm \sqrt{(U^2+V^2){\nu }^2-{\mu }^2{V}^2}]/(U^2+V^2)$. Reality of the square root does not guarantee positivity. For example, $U=1,V=0,\mu =-1,\nu =1/2$ gives two negative roots. Under the usual supercritical assumptions $\mu >0,U>0$, the plus root is positive whenever the discriminant is nonnegative.

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