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

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 $n\geq 3$, define (on the punctured space $U=\mathbb{R}^{n}\setminus\{0\}$) the function $f(x)=\left\|x\right\|^{2-n}$. Then

 $\frac{\partial f}{\partial x_{i}}=(2-n)\frac{x_{i}}{\left\|x\right\|^{n}},$

and

 $\frac{\partial^{2}f}{{\partial x_{i}}^{2}}=n(n-2)\frac{x_{i}^{2}}{\left\|x% \right\|^{n+2}}-(n-2)\frac{1}{\left\|x\right\|^{n}}$

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

Title examples of harmonic functions on $\mathbb{R}^{n}$ ExamplesOfHarmonicFunctionsOnmathbbRn 2013-03-22 12:44:23 2013-03-22 12:44:23 mathwizard (128) mathwizard (128) 9 mathwizard (128) Example msc 31A05 msc 31B05