= Solution
The <Fourier orthogonality> relation on this angular interval is
$$
\int_0^{2\pi\alpha}\sin\frac{n\theta}{2\alpha}\sin\frac{m\theta}{2\alpha}\,d\theta
=\pi\alpha\,\delta_{nm}.
$$
Projecting the outer <boundary condition> onto each angular <eigenfunction> gives
$$
\boxed{A_n=\frac{R^{-n/(2\alpha)}}{\pi\alpha}\int_0^{2\pi\alpha}\varphi(\theta)\sin\frac{n\theta}{2\alpha}\,d\theta.}
$$
The solution is \b[unique] in the class of bounded <harmonic functions> with the prescribed boundary trace. For continuous boundary data, the difference of two solutions is zero on the boundary and bounded at the vertex, so the <maximum principle for harmonic functions> makes it zero throughout the sector. Equivalently all its <Fourier sine series> coefficients vanish. The usual convergence interpretation applies if the data are only piecewise smooth.
Back to article page
