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
Or it can be extended to two variables as
 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
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 to ) 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
If the value required to be discriminated is away from zero, , where is the target value, is used.
The interesting thing about this particular delta is its frequency domain representation: if we take as the Fourier transform of a Dirac delta function, we have . 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 , where represents time and represents the wave's frequency in hertz; the approximation is done by making the frequency as ludicrously high as possible.
 Working with integration
The way the Dirac delta function is used in integration is given by the following identity:
 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 .
If denotes the space of infinitely differentiable functions on the real line which are 0 outside of some bounded set, one can consider the space of linear functionals which are continuous in some appropriate sense. The set of ordinary functions can be regarded as a subset of , by identifying a function f with the map defined by . Thus the space can be thought of as an enlargement of the space of functions. The delta "function" is another element of this space, defined by , and so it's sometimes called a "generalized function".
An element is called a distribution on , 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.
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