= Solution
{c}
The boundary data consist of the third angular <Fourier mode>. Its regular radial factor in the <harmonic expansion on a disk> is $r^3$, so
$$
\boxed{T(r,\theta)=r^3\sin(3\theta)}.
$$
For a direct check of <Laplace's equation>,
$$
T_{rr}+\frac1rT_r+\frac1{r^2}T_{\theta\theta}
=6r\sin(3\theta)+3r\sin(3\theta)-9r\sin(3\theta)=0.
$$
Also $T(1,\theta)=\sin(3\theta)$. At the origin the apparent coordinate singularity is harmless: this is the <harmonic polynomial> $3x^2y-y^3$. The allowed uniqueness for the <Dirichlet problem> completes the identification.
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