= Separated polar solutions of the biharmonic equation
{title2=$\Delta^2(r^m e^{in\theta})=(m^2-n^2)((m-2)^2-n^2)r^{m-4}e^{in\theta}$}
For angular <Fourier modes> of integer order $n\geq2$, the <biharmonic equation> in <polar coordinates> has radial basis $r^n,r^{-n},r^{n+2},r^{2-n}$. Apply the <Laplacian> twice to obtain the displayed indicial equation. For $n=0$, repeated roots give the radial basis $1,\log r,r^2,r^2\log r$; for $n=1$, the basis is $r^{-1},r,r\log r,r^3$. These are <Cauchy-Euler equations>. Regularity at the origin and <boundary conditions> select the admissible powers. For $n=2$, a constant radial factor still multiplies $\cos2\theta$ or $\sin2\theta$, so it is bounded but fails continuity at the origin.
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