ExamplesOfHarmonicFunctionOnRn 2002-06-05 05:17:38 2004-08-09 06:49:27 Example harmonic function % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} \newcommand{\norm}[1]{\left\|#1\right\|} 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? \begin{itemize} \item 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 $\Delta f\equiv 0$. \item 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~$\Delta f\equiv 0$. \item 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. \end{itemize}