Dirac Delta Function

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The Dirac delta function is an important mathematical tool that comes in handy in different fields.

In one dimension, the "delta" function $\delta(x)\,$ is defined as zero if $x\ne 0\,$ and is infinite at $x = 0$ . More accurately, it is a pulse of no width but unit area. Even more accurately, it is the limit as a unit area pulse of any shape gets ever narrower but at the same time taller, keeping a constant area. Really pedantic mathematicians refuse to call it a function, instead referring to the "delta symbol".

In applications to the real world, the delta function is indispensible for representeing so-called "point" sources and is a valuable tool in signals and systems theory as well as many other branches of engineering and physics.

1 Definitions
2 Properties of the Delta Function
- 2.1 Fourier transform
- 2.2 Convolution and Multiplication by Delta
3 Relatives of the Delta Function
- 3.1 Shah
- 3.2 Impulse Pairs

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Definitions

Symbolically, $δ$ is defined so that: $1 = \int_{-\infty}^\infty \delta(x)\, dx$

and

$f(x) = \int_{-\infty}^\infty f(x^\prime)\delta(x-x^\prime)\, dx$ .

In multiple dimensions, we can also define a delta function. However, we must be careful to account for the surface or volume elements correctly. For example, in rectangular coordinates, we can simply write

$\delta(\textbf{r}-\textbf{r}^\prime) = \delta(x-x^\prime)\delta(y-y^\prime)\delta(z-z^\prime)\,$ .

In circular cylindrical coordinates, it becomes a little more tricky, namely:

$\delta(\textbf{r}-\textbf{r}^\prime) = \frac{1}{\pi r} \delta(r-r^\prime)\delta(\phi-\phi^\prime)\delta(z-z^\prime)\,$ ,

except when the primed radial coordinate $r^\prime=0\,$ when the function becomes

$\delta(\textbf{r}) = \frac{1}{\pi r} \delta(r)\delta(z-z^\prime)\,$ ,

This is a consequence of

$1=\int\!\int\!\int \delta(\textbf{r}-\textbf{r}^\prime) \,r\, dr\, d\phi\, dz$ ,

where the integration volume must enclose the point $\textbf{r}^\prime\,$

Likewise, in spherical coordinates, we have

$\delta(\textbf{r}-\textbf{r}^\prime) = \frac{1}{4\pi r^2} \delta(r-r^\prime)\delta(\theta-\theta^\prime) \delta(\phi-\phi^\prime)\,$ ,

and when the primed radial coordinate vanishes, one gets

$\delta(\textbf{r}) = \frac{1}{4\pi r^2} \delta(r) \,$ ,

as a result of

$1=\int\!\int\!\int \delta(\textbf{r}-\textbf{r}^\prime) \, r^2\sin\theta\, dr\, d\theta\, d\phi$ .

Again, the integration volume must enclose the point $\textbf{r}^\prime\,$

For further information, check out the Delta function page on the Mathworld site.

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Properties of the Delta Function

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Fourier transform

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Convolution and Multiplication by Delta

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Relatives of the Delta Function

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Shah

The shah function represents an infinite train of delta pulses. $I I I (x)$ has a delta on every whole value of $x$ .

Shah has some very important properties:

It is its own Fourier (see Fourier Analysis) transform.
An instrument sampling a signal at regular intervals can be represented by a multiplication by shah.
A signal repeating forever is the convolution of that signal with shah.

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