Properties of the Dirac Delta Function

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Section 8.3 Properties of the Dirac Delta Function

There are many properties of the delta function which follow from the defining properties in Section 8.2. 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 $a=\hbox{constant}$ and $g(x_i) = 0\text{,}$ $g'(x_i) \ne 0\text{.}$ The first two properties show that the delta function is even and its derivative is odd.

Question 8.3.1. Prove properties of the delta function.

Prove some or all of the properties of the Dirac delta function listed in Section 8.3.