Properties of the Dirac Delta Function

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Section 11.3 Properties of the Dirac Delta Function

There are many properties of the delta function which follow from the defining properties in Section 11.2. 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 11.3.1. Prove properties of the delta function.

Prove some or all of the properties of the Dirac delta function listed in Section 11.3.