Power-law angular mode for the planar Poisson equation ($\Delta [r^s\cos (m\theta )]=(s^2-m^2)r^{s-2}\cos (m\theta )$)

For integer angular frequency $m\ge 0$, the Laplacian in polar coordinates gives the displayed identity on $r>0$ by direct differentiation. Consequently a source $\beta r^q\cos (m\theta )$ has particular solution $\beta r^{q+2}\cos (m\theta )/[(q+2{)}^2-m^2]$ when the denominator is nonzero. The sine case is identical. If $s=q+2=\pm m\ne 0$, differentiating the identity with respect to $s$ gives the resonant particular solution $\beta r^s\ln r\cos (m\theta )/(2s)$. For $s=m=0$, the radial source $\beta r^{-2}$ instead has particular solution $\beta (\ln r{)}^2/2$. These are local statements on the punctured plane; boundedness, smoothness at the origin and boundary data must be imposed separately by adding suitable harmonic functions.