# construction of Dirac delta function

The Dirac delta function is notorious in mathematical circles for having no actual 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^{} $\epsilon $ and define the hyperreal valued function $\delta {:}^{*}\mathbb{R}{\u27f6}^{*}\mathbb{R}$ by

$$ |

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,

$${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\delta (x)\mathit{d}x={\int}_{-\epsilon /2}^{\epsilon /2}\frac{1}{\epsilon}\mathit{d}x=1,$$ |

so the integral property is satisfied. Finally, for any continuous^{} real function $f:\mathbb{R}\u27f6\mathbb{R}$, choose an infinitesimal $z>0$ such that $$ for all $$; then

$$ |

which implies that ${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\delta (x)f(x)\mathit{d}x$ is within an infinitesimal of $f(0)$, and thus has real part equal to $f(0)$.

Title | construction of Dirac delta function |
---|---|

Canonical name | ConstructionOfDiracDeltaFunction |

Date of creation | 2013-03-22 12:35:48 |

Last modified on | 2013-03-22 12:35:48 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Derivation^{} |

Classification | msc 34L40 |

Classification | msc 26E35 |