# Delta function

 $y = \sin(x)\,$ $\frac{dy}{dx}=\ ?$ This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

Delta function is a Godsend to students as a way to discriminate numbers; it usually comes in two flavours.

## Kronecker's Delta function

Also known as the unit impulse, it is defined as a function that has value of 1 if the variables are 0, and 0 if they are not. As a single variable function, it is usually written as

$\delta_{i} = \begin{cases} 1, & i = 0 \\ 0, & i \ne 0 \end{cases}$

Or it can be extended to two variables as

$\delta_{ij} = \left\{\begin{matrix} 1, & \mbox{if } i=j \\ 0, & \mbox{if } i \ne j \end{matrix}\right.$

## Dirac Delta function

Paul Dirac says, "You can't integrate that and end up with meaningful stuff!" and goes off to invent a function that works with integration. It is defined as

$\delta(x) = \lim_{a \to 0}\frac{1}{a \sqrt{\pi}} \mathrm{e}^{-x^2/a^2}$

At the limit, it has a nonzero value if the variables is 0, and 0 if they are not. Since the total area over the real line (from $-\infty$ to $\infty$) is defined to be 1, the value at 0 would be infinite. That gives the nickname of infinite impulse and it is usually written as

$\delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$

If the value required to be discriminated is away from zero, $\delta(x-y)$, where $y$ is the target value, is used.

The interesting thing about this particular delta is its frequency domain representation: if we take $F(\omega)$ as the Fourier transform of a Dirac delta function, we have $F(\omega) = 1$. This means the following things:

• A Dirac pulse function has constant amplitude in the entire frequency spectrum. This means we must be careful when transmitting a delta function over a transmission channel, because it will interfere with every single other transmission made in the medium's passband.
• Dirac pulses have infinite energy, because their bandwidth is infinite. Therefore, they're impossible to transmit on a real system - to do so, the normal practice is to approximate the Dirac pulse with the cardinal sine function $sinc(2 \pi f T) = sin(2 \pi f T) / 2 \pi f T$, where $T$ represents time and $f$ represents the wave's frequency in hertz; the approximation is done by making the frequency $f$ as ludicrously high as possible.

### Working with integration

The way the Dirac delta function is used in integration is given by the following identity:

$\int_{a}^b f(x) \, \delta(x-y)dx = f(y)$ iff $a

### What is it, really?

Strictly speaking, the delta function isn't actually a function at all: a function must have a real value at every point, and delta doesn't. What it is is a "generalized function" or "distribution": it's a gadget that when fed a function, spits out a real number, in this case $f(0)$.

If $\mathcal D(\mathbb R)$ denotes the space of infinitely differentiable functions on the real line which are 0 outside of some bounded set, one can consider the space $\mathcal D^\prime(\mathbb R)$ of linear functionals $T : \mathcal D(\mathbb R) \to \mathbb R$ which are continuous in some appropriate sense. The set of ordinary functions can be regarded as a subset of $\mathcal D^\prime(\mathbb R)$, by identifying a function f with the map $T_f \in \mathcal D^\prime(\mathbb R)$ defined by $T_f(g) = \int_{-\infty}^\infty fg \, dx$. Thus the space $\mathcal D^\prime(\mathbb R)$ can be thought of as an enlargement of the space of functions. The delta "function" is another element of this space, defined by $\delta(g) = g(0)$, and so it's sometimes called a "generalized function".

An element $\mathcal D^\prime(\mathbb R)$ is called a distribution on $\mathbb R$, and it turns out that it's possible to carry out many familiar operations on these generalized functions. For example, the delta function can be regarded in a very rigorous sense as being the derivative of the unit step function! Fourier analysis too carries over to the space of distributions largely intact.

Mathematics Articles on RationalWiki

Conservapedian mathematics  -  Fermat's last theorem  -  Fibonacci sequence  -  Golden Ratio  -  Groups  -  Gödel's incompleteness theorems  -  Hypatia of Alexandria  -  Information  -  Mathematics  -  Metric system  -  Phli (fun)  -  Pyramid  -  Rene Descartes  -  Sophie Germain  -  Statistics  -  wikiFactor  -  Zero  -