= Solution
{c}
A temperature on the full unit <disk> must be single valued, so its angular factor and angular <derivative> are $2\pi$-periodic. For positive <separation constant> $\lambda=k^2$, this requires $k=n$ to be a positive integer. For $\lambda=0$, <periodicity> removes $D\theta$. For $\lambda<0$, no nonzero linear combination of $e^{k\theta}$ and $e^{-k\theta}$ is $2\pi$-periodic, so that case disappears.
The origin belongs to the <disk>, so a classical <harmonic function> must remain bounded and extend smoothly there. This excludes $r^{-n}$ and $\log r$. The remaining individual separated solutions are constants and $r^n(A_n\cos(n\theta)+B_n\sin(n\theta))$. By <superposition>, the regular <harmonic expansion on a disk> is
$$
\boxed{T(r,\theta)=A_0+\sum_{n=1}^{\infty}r^n
\bigl(A_n\cos(n\theta)+B_n\sin(n\theta)\bigr)}.
$$
The admissible <Fourier coefficients> are fixed by the <boundary conditions>. Smoothness at the origin can also be seen from $r^n\cos(n\theta)$ and $r^n\sin(n\theta)$ being the real and imaginary parts of $(x+iy)^n$, hence <harmonic polynomials>.
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