Modified Helmholtz adjoint plane wave
= Modified Helmholtz adjoint plane wave
{title2=$v_\lambda$}
For $k>0$, a nonzero <spectral parameter for a linear boundary value problem> gives
$$
v_\lambda=\exp[ik(\lambda z-\overline z/\lambda)/2]=e^{Ax+By},\quad A=ik(\lambda-\lambda^{-1})/2,\quad B=-k(\lambda+\lambda^{-1})/2.
$$
The identity $A^2+B^2=k^2$ proves that this solves the <modified Helmholtz equation>. The reciprocal companion is $v_{1/\lambda}=\overline{v_{\overline\lambda}}$.