Physical-sheet growth rate of a quartic impulse response (${\sigma }_{\mathrm{abs}}=-\mathrm{Re}{a}_3+\max \{0,\mathrm{Re}(a_1^2/(4a_2))\text{ on the physical sheet}\}$)

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 $\mathrm{Re}z>0$; 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 $-\mathrm{Re}{a}_3+\max (\mathrm{Re}{a}_1,0{)}^2/(4\mathrm{Re}{a}_2)$ to distinguish absolute wave-packet instability from convective wave-packet instability. Equality cases require their algebraic prefactors.