Dirac Delta Distribution

Subsection A.3.1 Dirac Delta Distribution

Here we define the Dirac delta distribution and its relation to the unit step function.

Definition A.3.1 Dirac delta distribution.

The Dirac delta distribution \(\delta(x)\)is a generalized distribution that equals zero everywhere except at the point \(x=0\) with a total area of one:

\begin{equation*} \delta(x) = \begin{cases} \infty & x=0 \\ 0 & x\neq 0 \end{cases} \end{equation*}

Note A.3.2.

Conventionally, the value of the integral of the Dirac delta distribution is unambiguous whenever the support of the delta function strictly lies inside or outside the bounds of the interval over which the integral is being taken. Ambiguity arises when the delta spike occurs precisely at the bounds of the interval.

For the purposes of There and Back Again, we will mathematically represent this nuance as follows:

\begin{equation*} \int^b_a {\delta(x-c) \, dx} = \begin{cases} 0 & \text{if } c\in(-\infty,a) \cup (b,\infty) \\ 1 & \text{if } c\in(a,b) \\ * & \text{if } c\in {a,b} \end{cases} \end{equation*}

full-contribution convention: \begin{equation*} * = 1 \end{equation*}
symmetric convention: \begin{equation*} * = \frac{1}{2} \end{equation*}

Induced by the symmetric Heaviside step function \(H(x)\) where \(H(0)=\tfrac{1}{2}\) .

Note A.3.3.

This is an example of a statement whose value depends on an interpretive convention rather than solely on formal manipulation. The distinction appears because the Dirac delta is a distribution, not an ordinary function.