Some real functions in $\mathbb{R}^n$ (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 $U=\mathbb{R}^n \setminus \{0\}$ ) the function $f(x)=\norm{x}^{2-n}$ . Then $$ \frac{\partial f}{\partial x_i} = (2-n) \frac{x_i}{\norm{x}^n}, $$ and $$ \frac{\partial^2 f}{{\partial x_i}^2} = n(n-2)\frac{x_i^2}{\norm{x}^{n+2}} - (n-2)\frac{1}{\norm{x}^n} $$ Summing over $i=1,...,n$ shows $\Delta f\equiv 0$ .
• For $n=2$ , define (on the punctured plane $U=\mathbb{R}^2 \setminus \{0\}$ ) the function $f(x,y)=\log(x^2+y^2)$ . Derivation and summing yield $\Delta f\equiv 0$ .
• For $n=1$ , the condition $(\Delta f)(x)=f''(x)\equiv 0$ forces $f$ to be an affine function on every segment; there are no ``interesting'' harmonic functions in one dimension.