  We show that a smooth radially symmetric solution $u$ to the graphic Willmore
surface equation is either a constant or the defining function of a half sphere
in ${\mathbb R}^3$. In particular, radially symmetric entire Willmore graphs in
${\mathbb R}^3$ must be flat. When $u$ is a smooth radial solution over a
punctured disk $D(\rho)\backslash\{0\}$ and is in $C^1(D(\rho))$, we show that
there exist a constant $\lambda$ and a function $\beta$ in $C^0(D(\rho))$ such
that $u''(r) =\frac{\lambda}{2}\log r+\beta(r)$; moreover, the graph of $u$ is
contained in a graphical region of an inverted catenoid which is uniquely
determined by $\lambda$ and $\beta(0)$. It is also shown that a radial solution
on the punctured disk extends to a $C^1$ function on the disk when the mean
curvature is square integrable.
