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\)
Section 16.8 Properties of the Dirac Delta Function
There are many properties of the delta function which follow from the defining properties in
Section 16.4 . Some of these are:
\begin{align*}
\delta(x) \amp = \delta(-x)\\
\frac{d}{dx}\,\delta(x) \amp = -\frac{d}{dx}\,\delta(-x)\\
\int_b^c f(x)\, \delta'(x-a)\, dx \amp = -f'(a)\\
\delta(ax) \amp = {1\over \vert a \vert}\,\delta(x)\\
\delta\bigl(g(x)\bigr)
\amp = \sum_i {1 \over \vert g'(x_i) \vert} \,\delta(x-x_i)\\
\delta(x^2-a^2) \amp =
\vert 2a \vert^{-1} \left[ \delta(x-a) + \delta(x+a)\right]\\
\delta\bigl( (x-a)(x-b) \bigr)
\amp = {1 \over \vert a-b \vert} \left[\delta(x-a)
+ \delta(x-b)\right]
\end{align*}
where \(a\) and \(b\) are real-valued constants and the function \(g(x)\) has zeros at \(x_i\) with properties \(g(x_i) = 0\) and \(g'(x_i) \ne 0\text{.}\) The first two properties show that the delta function is even and its derivative is odd.
Sensemaking 16.3 . Prove properties of the delta function.
Prove some or all of the properties of the Dirac delta function listed in
Section 16.8 .
