Properties of the Dirac Delta Function

There are many properties of the delta function which follow from the defining properties in § {The Dirac Delta Function}. Some of these are: \begin{eqnarray} \delta(x) &=& \delta(-x) \\ \frac{d}{dx}\,\delta(x) &=& -\frac{d}{dx}\,\delta(-x) \\ \int_b^c f(x)\, \delta'(x-a)\, dx &=& -f'(a) \\ \delta(ax) &=& {1\over \vert a \vert}\,\delta(x) \\ \delta\bigl(g(x)\bigr) &=& \sum_i {1 \over \vert g'(x_i) \vert} \,\delta(x-x_i) \\ \delta(x^2-a^2) &=& \vert 2a \vert^{-1} \left[ \delta(x-a) + \delta(x+a)\right] \\ \delta\bigl( (x-a)(x-b) \bigr) &=& {1 \over \vert a-b \vert} \left[\delta(x-a) + \delta(x-b)\right] \end{eqnarray} where $a=\hbox{constant}$ and $g(x_i) = 0$, $g'(x_i) \ne 0$. The first two properties show that the delta function is even and its derivative is odd.

Personal Tools