= Rational-angle oblique derivative problem on a quadrant
{title2=$\beta_1+\beta_2=n\pi/2$}
For a complex-valued <harmonic function> in the first quadrant, write $\Phi=q_z$ and $\Psi(\bar z)=q_{\bar z}$ as independent <holomorphic functions>. Prescribe $i(e^{i\beta_1}\Phi-e^{-i\beta_1}\Psi)=g_1$ on the imaginary axis and $e^{-i\beta_2}\Phi+e^{i\beta_2}\Psi=g_2$ on the real axis. If $\beta_1+\beta_2=n\pi/2$, set $\epsilon=(-1)^n$. Reflection into four quadrants produces a scalar <Riemann-Hilbert problem> with parity $H(-z)=-\epsilon H(z)$ and known jumps. Provided its corner and large-arc contributions vanish, the <Cauchy integral formula> gives
$$
\Phi(z)=\frac1{2\pi i}\left[e^{i\beta_2}\int_0^\infty g_2(s)\left(\frac1{s-z}-\frac\epsilon{s+z}\right)ds-e^{-i\beta_1}\int_0^\infty g_1(s)\left(\frac1{is-z}-\frac\epsilon{is+z}\right)ds\right].
$$
Complex boundary data do not justify replacing $q_{\bar z}$ by the conjugate of $q_z$.
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