Section 11.3 Exercise: Delta Functions 1
¶Prove that the derivative of the step function is the delta function, i.e.~show that \begin{equation} \frac{d}{dx}\,\Theta(x-a) \end{equation} satisfies the property of the delta function in equation~(\ref{fdelta}) in Section~\ref{deltaintro}.
We need to show that
\begin{equation}
\int_{b}^{c} f(x)\,\delta(x-a) \,dx
= f(a)\tag{11.3.1}
\end{equation}
for \(b\lt a\lt c\text{,}\) where
\begin{equation}
\delta(x-a)=\frac{d}{dx}\,\Theta(x-a)\tag{11.3.2}
\end{equation}
The main strategy is to use integration by parts, paying strict attention to the limits of integration.
\begin{align}
\int_{b}^{c} f(x)\,\delta(x-a) \,dx
\amp= \int_{b}^{c} f(x)\,\frac{d}{dx}\, \Theta(x-a) \,dx\notag\\
\amp= \left. f(x)\, \Theta(x-a)\right|_b^c
- \int_{b}^{c} \frac{d}{dx}(f(x))\,\Theta(x-a) \,dx\notag\\
\amp= \left\{ f( c )-0\right\}
- \int_{a}^{c} \frac{d}{dx}(f(x))\,dx\notag\\
\amp= f( c )-\left\{\left. f(x)\right|_a^c\right\}\notag\\
\amp= f( c )-\left\{f( c )-f(a)\right\}\notag\\
\amp= f(a)\tag{11.3.3}
\end{align}