= Solution
Regularity at $r=0$ excludes every negative radial power and the logarithmic zero mode. Thus the general regular single-valued <harmonic function> on the <disk> has the form
$$
\boxed{\phi(r,\theta)=A_0+\sum_{n=1}^{\infty}r^n(A_n\cos n\theta+B_n\sin n\theta).}
$$
The coefficients must give a <series> converging locally in the disk; if finiteness and continuity on its boundary are required, the boundary <Fourier series> must represent the prescribed finite continuous boundary values. Conversely every such locally convergent expansion is <harmonic>, and the <Fourier coefficients> on circles show that it includes every regular solution. There is no angular term linear in $\theta$, because it would be multivalued.
Back to article page
