Proposition 12.4.17.label A radial function $f(x)$ is harmonic in $\R^{n}\del\curl{0}\iff \begin{cases}\p(r)=a+br^{2-n}&n\geq 3\\ \p(r)=a+b\log r&n=2\end{cases}$
Proof. By Proposition 12.4.14 we have
\begin{align*}\la f=0\iff\frac{\p''}{\p'}=-\frac{n-1}{r}=\frac{1-n}{r}\end{align*}
the logarithmic derivative. Thus
\begin{align*}\log\p'=(1-n)\log r+const=\log(r^{1-n}\cd const)\end{align*}
So $\p'=const\cd r^{1-n}$ then integration finishes the proof.$\square$
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