= Rotation-invariant Kähler potential on the complex plane
{title2=$u(t)=f(e^{t/2}),\quad t=\log|z|^2$}
A smooth real rotation-invariant potential is constant on every positive-radius circle, so its logarithmic-radius function $u$ is smooth on $\mathbb R$. Off the origin, $f_{z\bar z}=e^{-t}u''(t)$. Since $f$ is smooth on the entire plane, this coefficient extends smoothly through the origin. A finite <Taylor expansion>, rather than a convergent series assumption, gives $f(z)=f(0)+b|z|^2+O(|z|^4)$.
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