Definition:Dirac Delta Function/Definition 2

Definition
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

Consider the real function $F_\epsilon: \R \to \R$ defined as:


 * $\map {F_\epsilon} x := \begin {cases}

0 & : x < -\epsilon \\ \dfrac 1 {2 \epsilon } & : -\epsilon \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$

The Dirac delta function is defined as:


 * $\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$

Which is equivalent to defining $\map \delta x$ by the following $2$ properties:

By Values of Dirac Delta Function over Reals
 * $\map \delta x := \begin {cases}

\infty & : x = 0 \\ 0 & : x \ne 0 \end {cases}$

and by Integral of Dirac Delta Function over Reals
 * $\ds \int_{-\infty}^{+\infty} \map \delta x \rd x = 1$

Also see

 * Equivalence of Definitions of Dirac Delta Function