examples of harmonic functions on ${ℝ}^n$

Some real functions in ${ℝ}^n$ (e.g. any linear function, or any affine function) are obviously harmonic functions. What are some more interesting harmonic functions?

• For $n\ge 3$, define (on the punctured space $U={ℝ}^n\setminus \{0\}$) the function $f(x)={\parallel x\parallel }^{2-n}$. Then

  $$\frac{\partial f}{\partial x_i}=(2-n)\frac{x_i}{{\parallel x\parallel }^n},$$

  and

  $$\frac{{\partial }^{2}f}{\partial x_i{}^2}=n(n-2)\frac{x_i^2}{{\parallel x\parallel }^{n+2}}-(n-2)\frac{1}{{\parallel x\parallel }^n}$$

  Summing over $i=1,\mathrm{\dots },n$ shows $\mathrm{\Delta }f\equiv 0$.

• For $n=2$, define (on the punctured plane $U={ℝ}^2\setminus \{0\}$) the function $f(x,y)=\log (x^2+y^2)$. Derivation and summing yield $\mathrm{\Delta }f\equiv 0$.

• For $n=1$, the condition $(\mathrm{\Delta }f)(x)=f^{\prime \prime }(x)\equiv 0$ forces $f$ to be an affine function on every segment; there are no “interesting” harmonic functions in one dimension.