 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 formThe 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.