= Harmonic extension in a circular sector
{title2=$\Phi(r,\theta)=\sum_{n\geq1}b_n(r/R)^{n\pi/\gamma}\sin(n\pi\theta/\gamma)$}
For $0<\theta<\gamma$ and $0<r<R$, a bounded <harmonic function> that vanishes on the straight edges and has outer boundary data $\varphi$ has coefficients
$$
b_n=\frac2\gamma\int_0^\gamma\varphi(\theta)\sin\frac{n\pi\theta}\gamma\,d\theta.
$$
This follows by <separation of variables> and <Fourier orthogonality>. Negative radial powers are excluded by boundedness at the vertex. Reflection-symmetric data $\varphi(\theta)=\varphi(\gamma-\theta)$ have only odd <Fourier modes>. Knowing the resulting solution on the bisector determines all these odd coefficients by its radial <power series>. Without reflection symmetry, the even angular modes vanish on the bisector and cannot be recovered from that restriction alone.
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