Properties of the Dirac delta function

1. Sifting property. Given function $\ f\,(x)$ continuous at $% x=x^{\prime }$,
   
   $$\int_{a}^{b}f\,(x^{\prime })\;\delta (x-x^{\prime })\,dx^{\p... ...f }(a,b)\mbox{ does not contain }x\mbox{ } \end{array}\right.$$
   
   
   When integrated, the product of any (well-behaved) function and the Dirac delta yields the function evaluated where the Dirac delta is singular. The sifting property also applies if the arguments are exchanged: $\ f\,(x^{\prime })\;\delta (x-x^{\prime })dx^{\prime }=f\,(x)\;\delta (x^{\prime }-x)dx$.

2. Integral.
   
   $$\int_{-\infty }^{t}\delta (\tau )\,d\tau =H(t);\frac{dH(t-\tau )}{dt}=\delta (t-\tau )$$
   
   
   where $H(t)$ is the Heaviside unit step function defined as
   
   $$H(t)=\left\{ \begin{array}{cc} 0 & \mbox{if \ }t<0 \\ 1 & \mbox{if \ }t>0 \end{array}\right.$$
   
   

3. Units. Since the definition of the Dirac delta requires that the product $\delta (x)dx$ is dimensionless, the units of the Dirac delta are the inverse of those of the argument $x$. That is, $\delta (x)$ has units $% meters^{-1}$, and $\delta (t)$ has units $sec^{-1}$.

4. Definition for radial, 2-D, and 3-D geometries. For two- and three- dimensional problems with vector coordinate $\overrightarrow{\mathbf{r}}$, the Dirac delta function is defined:
   
   $$\delta (\overrightarrow{\mathbf{r}})=0\mbox{ if }\overrighta... ...{-\infty }^{\infty }\delta (\overrightarrow{\mathbf{r}})\,dv=1$$
   
   
   where $dv$ is differential volume. The units of $\delta (\overrightarrow{% \mathbf{r}})$ are given by [$dv$]$^{-1}$, and three important cases are the listed below.
   1. 1-D radial cylindrical coordinates: $dv=2\pi rdr$, and units of $% \delta (\overrightarrow{\mathbf{r}})$ are [meters]$^{-2}$.

   2. 1-D radial spherical coordinates: $dv=4\pi r^{2}dr,$and units of $% \delta (\overrightarrow{\mathbf{r}})$ are [meters]$^{-3}$.

   3. 2-D Cartesian coordinates: dv = dx dy, and units of $\delta (% \overrightarrow{\mathbf{r}})$ are [meters]$^{-2}$.