Quarter-plane oblique boundary spectral functions (source code)

= Quarter-plane oblique boundary spectral functions
{title2=$\widehat q_1=\tfrac12e^{-i\beta_1}(H_1-iJ_1),\quad\widehat q_2=-\tfrac12e^{i\beta_2}(H_2-iJ_2)$}

For real <harmonic> $q$, prescribe $-\sin\beta_1q_x+\cos\beta_1q_y=h_1$ on the vertical axis and $\cos\beta_2q_x-\sin\beta_2q_y=h_2$ on the horizontal axis. Their orthogonal derivatives $j_1,j_2$ determine the missing gradient components. If $H_1,J_1$ use the kernel $e^{ky}$ and $H_2,J_2$ use $e^{-ikx}$, the clockwise spectra $\widehat q_1=\int_0^{i\infty}e^{-ikz}q_z\,dz$, $\widehat q_2=-\int_0^\infty e^{-ikz}q_z\,dz$ have the displayed expressions. Their <global relation> is $\widehat q_1+\widehat q_2=0$ in the third spectral quadrant.

// Target: analysis.bigb