Let \(u \in C^2(\mathbb{R}^d \setminus \{0\})\) be harmonic . Then
\[ v(x) = \frac{1}{|x|^{d-2}} u\left(\frac{x}{|x|^2}\right) \]is harmonic.
Proof
Apply the Laplacian in polar coordinates on \(v\).
Let \(u \in C^2(\mathbb{R}^d \setminus \{0\})\) be harmonic . Then
\[ v(x) = \frac{1}{|x|^{d-2}} u\left(\frac{x}{|x|^2}\right) \]is harmonic.