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Apply the formula toThe upper-half-plane poles areand their residues are $e^{i{z}_k}/(4z_k^2)$. Writing $c=1/\sqrt{2}$, their sum isSince the cosine part of $x{e}^{ix}/(1+x^4)$ is odd, its integral vanishes, while the sine part is even. Henceso

The parity of a wavefunction is its eigenvalue under spatial reflection: $\psi (-x)=+\psi (x)$ for even parity and $\psi (-x)=-\psi (x)$ for odd parity. If $V(-x)=V(x)$, the Hamiltonian commutes with reflection, so its energy eigenspaces admit a parity eigenbasis. In particular, every nondegenerate bound-state energy eigenfunction has definite parity; within a degenerate scattering eigenspace one must choose the even and odd linear combinations.

If the initial field is also an even function, then $A(k)$ is real and even. Provided $A(k_0)\ne 0$, the leading term becomesIts large zeros satisfyThereforeIf $A(k_0)=0$, the leading stationary-phase coefficient vanishes and a higher-order term determines the zeros.

Put $q=|k|$. Decay at infinity and the symmetry of the disturbance suggest the mode potentialsThe middle potential is an even function of $y$ because its vertical derivative must take opposite values at the two interfaces. The upper kinematic conditions giveThe lower conditions give the same two relations. The upper dynamic condition isEliminating $A$, $B$ and ${\eta }_0$ yields the dispersion relationWriting $C=\coth (qh)$, its two roots areFor every real $k\ne 0$, $C>0$ and one root has the positive growth rateThus every nonzero wavenumber is linearly unstable in this inviscid vortex sheet model; $k=0$ is neutral.