ConstructionOfDiracDeltaFunction 2002-04-19 01:04:11 2002-04-21 17:44:42 Derivation Dirac delta 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{\R}{\mathbb{R}} The Dirac delta function is notorious in mathematical circles for having no actual \PMlinkescapetext{realization} as a function. However, a little known secret is that in the domain of nonstandard analysis, the Dirac delta function admits a completely legitimate construction as an actual function. We give this construction here. Choose any positive infinitesimal $\varepsilon$ and define the hyperreal valued function $\delta:\,^*\R \longrightarrow\,^*\R$ by $$ \delta(x) := \begin{cases} 1/\varepsilon & -\varepsilon/2 < x < \varepsilon/2, \\ 0 & \text{otherwise.} \end{cases} $$ We verify that the above function satisfies the required properties of the Dirac delta function. By definition, $\delta(x) = 0$ for all nonzero real numbers $x$. Moreover, $$ \int_{-\infty}^\infty \delta(x)\ dx = \int_{-\varepsilon/2}^{\varepsilon/2} \frac{1}{\varepsilon} \ dx = 1, $$ so the integral property is satisfied. Finally, for any {\em continuous} real function $f: \R \longrightarrow \R$, choose an infinitesimal $z > 0$ such that $|f(x) - f(0)| < z$ for all $|x| < \varepsilon/2$; then $$ \varepsilon \cdot \frac{f(0) - z}{\varepsilon} < \int_{-\infty}^\infty \delta(x) f(x)\ dx < \varepsilon \cdot \frac{f(0) + z}{\varepsilon} $$ which implies that $\int_{-\infty}^\infty \delta(x) f(x)\ dx$ is within an infinitesimal of $f(0)$, and thus has real part equal to $f(0)$.