= Solution
Select the <causal Green function>, zero for $t<0$; otherwise the impulse equation permits arbitrary homogeneous additions. Use the spatial <Fourier transform> convention $e^{-ikx}$ and the temporal <Laplace transform> boundary convention $e^{i\omega t}$. The transformed impulse equation is
$$
\widehat G(k,\omega)=\frac1{-i\omega+iUk-\mu+\gamma k^2}.
$$
If $\mu>0$, a literal real-frequency <Fourier transform> of the growing response need not exist. Its causal inverse instead starts on $\operatorname{Im}\omega=c>\max(\mu,0)$ and is continued from there. For $t>0$ close the frequency contour below; its clockwise orientation and the <pole> <residue> $1/(-i)=i$ give $e^{-i\omega(k)t}$. For $t<0$ the upper closure contains no <pole>. The remaining inverse spatial transform is a <Gaussian integral>:
$$
G(x,t)=\frac{H(t)e^{\mu t}}{2\pi}\int_{\mathbb R}
e^{-\gamma tk^2+ik(x-Ut)}\,dk
=\boxed{\frac{H(t)}{\sqrt{4\pi\gamma t}}
\exp\left[\mu t-\frac{(x-Ut)^2}{4\gamma t}\right]}.
$$
The unit-mass <Gaussian heat kernel> tends to $\delta(x)$ at $t\downarrow0$, giving the required impulse normalization.
At the advected center $x=Ut$, its <amplitude> is $e^{\mu t}/\sqrt{4\pi\gamma t}$. At a fixed spatial point,
$$
G(x,t)=\frac{e^{Ux/(2\gamma)}}{\sqrt{4\pi\gamma t}}
\exp\left[\left(\mu-\frac{U^2}{4\gamma}\right)t-\frac{x^2}{4\gamma t}\right].
$$
Thus $\mu>0$ permits growth of the moving <wave packet>, but it is <convective hydrodynamic instability> if the fixed-position rate is negative. A positive fixed-position rate is <absolute hydrodynamic instability>. At its equality threshold the $t^{-1/2}$ factor decays, explaining the marginal exponential classification. This directly verifies the physically selected <Briggs-Bers criterion> saddle rather than assuming its accessibility.
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