 Rational-angle oblique derivative problem on a quadrant

ID: rational-angle-oblique-derivative-problem-on-a-quadrant

For a complex-valued harmonic function in the first quadrant, write

Φ = q_{z}

and

Ψ (\overset{z}{ˉ}) = q_{\overset{z}{ˉ}}

as independent holomorphic functions. Prescribe

i (e^{i β_{1}} Φ - e^{- i β_{1}} Ψ) = g_{1}

e^{- i β_{2}} Φ + e^{i β_{2}} Ψ = g_{2}

on the real axis. If

β_{1} + β_{2} = nπ /2

ϵ = (- 1)^{n}

. Reflection into four quadrants produces a scalar Riemann-Hilbert problem with parity

H (- z) = - ϵH (z)

and known jumps. Provided its corner and large-arc contributions vanish, the Cauchy integral formula gives

Φ (z) = \frac{1}{2 πi} [e^{i β_{2}} \int_{0}^{\infty} g_{2} (s) (\frac{1}{s - z} - \frac{ϵ}{s + z}) d s - e^{- i β_{1}} \int_{0}^{\infty} g_{1} (s) (\frac{1}{i s - z} - \frac{ϵ}{i s + z}) d s] .

Complex boundary data do not justify replacing

q_{\overset{z}{ˉ}}

by the conjugate of

q_{z}

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