= Solution
For $r>0$, the <chain rule> gives $r_x=x/r$ and $r_y=y/r$. Applying it twice to the <radial function> $u=f(r)$ gives
$$
u_{xx}=f''\frac{x^2}{r^2}+f'\left(\frac1r-\frac{x^2}{r^3}\right),\qquad u_{yy}=f''\frac{y^2}{r^2}+f'\left(\frac1r-\frac{y^2}{r^3}\right).
$$
Using $x^2+y^2=r^2$, <Laplace's equation> becomes
$$
f''+\frac1rf'=0,\qquad (rf')'=0.
$$
Two <integrations> now give
$$
\boxed{u(x,y)=A\log r+B,\qquad r=\sqrt{x^2+y^2}>0.}
$$
These are the general solutions of the specified <radial function> form on an <annulus> or a punctured plane. If the solution must extend continuously to the origin, the <logarithm> term forces $A=0$, leaving only a constant <function>.
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