= Solution
Take the physical coefficients $U,\mu,c_d$ real. Substitution of $e^{i(kx-\omega t)}$ into the <linear complex Ginzburg-Landau equation> gives $-i\omega+ikU-\mu+(1+ic_d)k^2=0$, hence
$$
\omega=Uk+(c_d-i)k^2+i\mu.
$$
Let $A=c_d-i$. <Completing the square> yields $\omega=A(k+U/(2A))^2+i\mu-U^2/(4A)$. Therefore the <absolute wavenumber> and <absolute frequency> are
$$
\boxed{k_0=-\frac{U}{2(c_d-i)},\qquad
\omega_0=i\mu-\frac{U^2}{4(c_d-i)}.}
$$
Their explicitly separated parts are
$$
k_0=-\frac{U(c_d+i)}{2(1+c_d^2)},\qquad
\omega_0=-\frac{U^2c_d}{4(1+c_d^2)}
+i\left[\mu-\frac{U^2}{4(1+c_d^2)}\right].
$$
Thus $d\omega/dk=U+2(c_d-i)k$ vanishes at $k_0$. The complex saddle describes the localized <Green function> response; it is different from selecting a real <wavenumber> for a <Fourier mode> for temporal amplification.
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