Quadrant modified Helmholtz spectral reconstruction ($q=(4\pi i{)}^{-1}{\int }_{{\ell }_x\cup {\ell }_y}{e}^{ikz+\lambda \bar{z}/(ik)}\rho (k)dk/k$)

For $\lambda >0$, put $a=k-\lambda /k$, $b=k+\lambda /k$. The spectral integral uses the positive real and positive imaginary rays, both oriented outward. On the first ray its density is ${\rho }_x=i{\int }_0^{\infty }{e}^{-iax}[bq(x,0)-q_y(x,0)]dx$; on the second it is ${\rho }_y={\int }_0^{\infty }{e}^{by}[aq(0,y)-i{q}_x(0,y)]dy$. The global relation ${\rho }_x+{\rho }_y=0$ holds in the third spectral quadrant. Primitives from the corner and from each infinite boundary end give an additive Riemann-Hilbert problem. Their limits are $-q$ at infinity and $q$ at zero, which explains the factor $1/(4\pi i)$.