= Power-law angular mode for the planar Poisson equation
{title2=$\Delta[r^s\cos(m\theta)]=(s^2-m^2)r^{s-2}\cos(m\theta)$}
For integer angular frequency $m\geq0$, 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 $s=q+2=\pm m\ne0$, 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>.
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