= Solution
Put $\Psi=U(r)V(\phi)$ in the <Laplace equation> in <polar coordinates>. After multiplication by $r^2/(UV)$, <separation of variables> gives
$$
\frac{r^2U''+rU'}U=-\frac{V''}V=n^2.
$$
The angular function is $2\pi$-periodic and single-valued, so its nonconstant modes are $\cos(n\phi),\sin(n\phi)$ with integer $n\geq1$. Their radial <Euler differential equation> has solutions $r^n,r^{-n}$; finiteness at the origin excludes $r^{-n}$.
The zero angular mode has constant angular part and radial solutions $1,\log r$. The logarithmic term is again excluded at the origin. <Superposition> therefore yields
$$
\boxed{\Psi(r,\phi)=A_0+\sum_{n=1}^\infty r^n[A_n\cos(n\phi)+B_n\sin(n\phi)]}.
$$
This is the regular harmonic expansion inside the disk.
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