= Quadrant modified Helmholtz spectral reconstruction
{title2=$q=(4\pi i)^{-1}\int_{\ell_x\cup\ell_y}e^{ikz+\lambda\bar z/(ik)}\rho(k)\,dk/k$}
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 $\rho_y=\int_0^\infty e^{by}[a q(0,y)-i q_x(0,y)]dy$. 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 $1/(4\pi i)$.
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