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Section 16.4 The Dirac Delta Function
Definition 16.9 . The Dirac Delta Function.
The
Dirac delta function \(\delta(x)\) has has the following defining properties:
\begin{equation}
\delta(x)
= \begin{cases}
0\quad \amp x\not= 0\\
\infty\quad\amp x=0
\end{cases}\tag{16.4.1}
\end{equation}
\begin{equation}
\int_b^c \delta(x)\, dx = 1 \qquad\qquad b\lt 0\lt c\tag{16.4.2}
\end{equation}
\begin{equation}
x\,\delta(x) \equiv 0\text{.}\tag{16.4.3}
\end{equation}
Using the Dirac Delta Function in an Integral.
The properties of the delta function allow us to compute
\begin{align*}
\int_{-\infty}^{\infty} f(x)\,\delta(x) \,dx
\amp = \int_{-\infty}^{\infty} f(0)\,\delta(x) \,dx\\
\amp = f(0) \int_{-\infty}^{\infty} \delta(x) \,dx\\
\amp = f(0)
\end{align*}
In the first equality, it is safe to replace the function \(f(x)\) with its value \(f(0)\) at \(x=0\) since everywhere else the integrand is zero, due to the delta function.
We can shift the “spike” in the delta function as usual, obtaining \(\delta(x-a)\text{.}\) This shifted delta function satisfies
\begin{equation}
\int_{-\infty}^{\infty} f(x)\,\delta(x-a) \,dx
= f(a)\tag{16.4.4}
\end{equation}
To Remember. The Dirac delta function can be used inside an integral to pick out the value of a function at any desired point.
