= Physical-sheet growth rate of a quartic impulse response
{title2=$\sigma_{\rm abs}=-\operatorname{Re}a_3+\max\{0,\operatorname{Re}(a_1^2/(4a_2))\text{ on the physical sheet}\}$}
Put $z=a_1/\sqrt{a_2}$ using the root with positive real part. The additional resolvent singularity at $s+a_3=z^2/4$ lies on the physical square-root sheet only when $\operatorname{Re}z>0$; otherwise the apparent saddle is on the other sheet. The singularity at $s=-a_3$ remains, including a quartic degeneracy when $a_1=0$. The rightmost physical singularity determines the exponential fixed-position growth rate. Compare it with the temporal maximum $-\operatorname{Re}a_3+\max(\operatorname{Re}a_1,0)^2/(4\operatorname{Re}a_2)$ to distinguish <absolute wave-packet instability> from <convective wave-packet instability>. Equality cases require their algebraic prefactors.
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