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THE GEOMETRY OF MATHEMATICAL METHODS

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.