Dirac delta function
Probability density function Schematic representation of the Dirac delta function for x_{0} = 0. A line with an arrow is usually used to schematically represent the Dirac delta function. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.  
Cumulative distribution function Using the halfmaximum convention, with x_{0} = 0  
Parameters  location (real) 
Support  
Probability density function (pdf)  
Cumulative distribution function (cdf)  (Heaviside) 
Mean  
Median  
Mode  
Variance  
Skewness  
Kurtosis  (undefined) 
Entropy  
mgf  
Char. func. 
The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. The integral from minus infinity to plus infinity is 1. The discrete analog of the delta "function" is the Kronecker delta which is sometimes known as a delta function. Note that the Dirac delta is not a function, but a distribution that is also a measure.
Overview 
Despite its name, the delta function is not a function. One reason for this is because the functions f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f, g are functions such that f = g almost everywhere, then f is integrable iff g is integrable and the integrals of f and g are the same. Rigorous treatment of the Dirac delta requires measure theory or the theory of distributions.
The Dirac delta is very useful as an approximation for a tall narrow spike function (an impulse). It is the same type of abstraction as a point charge, point mass or electron point. For example, in calculating the dynamics of a baseball being hit by a bat, approximating the force of the bat hitting the baseball by a delta function is a helpful trick. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball.
The Dirac delta function was named after the Kronecker delta, since it can be used as a continuous analogue of the discrete Kronecker delta.
Definitions
The Dirac delta function can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,
and which is also constrained to satisfy the identity
This heuristic definition should not be taken too seriously though. Firstly, the Dirac delta is not a function, as no function has the above properties. Moreover there exist descriptions which differ from the above conceptualization. For example, sinc(x / a) / a (where sinc is the sinc function) behaves as a delta function in the limit of , yet this function does not approach zero for values of x outside the origin.
The defining characteristic
where f is a suitable test function, cannot be achieved by any function, but the Dirac delta function can be rigorously defined either as a distribution or as a measure.
In terms of dimensional analysis, this definition of δ(x) implies that δ(x) has dimensions reciprocal to those of dx.
The delta function as a measure
As a measure, δ(A) = 1 if , and δ(A) = 0 otherwise. Then,
for all continuous f.
As distributions, the Heaviside step function is an antiderivative of the Dirac delta distribution.
The delta function as a probability density function
As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by
for every test function . It is a distribution with compact support (the support being {0}). Because of this definition, and the absence of a true function with the delta function's properties, it is important to realize the above integral notation is simply a notational convenience, and not a true integral.
Thus, the Dirac delta function may be interpreted as a probability density function. Its characteristic function is then just unity, as is the moment generating function, so that all moments are zero. The cumulative distribution function is the Heaviside step function.
Equivalently, one may define as a distribution δ(ξ) whose indefinite integral is the function
usually called the Heaviside step function or commonly the unit step function. That is, it satisfies the integral equation
for all real numbers x.
Delta function of more complicated arguments
A helpful identity is the scaling property:
and so
This concept may be generalized to:
where x_{i} are the roots of g(x). In the integral form it is equivalent to
In an ndimensional space with position vector , this is generalized to:
where the integral on the right is over , the n1 dimensional surface defined by .
The integral of the timeshifted Dirac delta is given by:
Thus, the delta function is said to "sift out" the function at the value , when integrated over all time.
Similarly, the convolution:
means that the effect of convolving with the timeshifted Dirac delta is to timeshift by the same amount.
Fourier transform
Using Fourier transforms, one has
and therefore:
which is a statement of the orthogonality property for the Fourier kernel.
Laplace transform
The direct Laplace transform of the delta function is:
as a curious identity using Euler's formula 2cos(as) = e ^{− ias} + e^{ias} we got the Laplace inverse transform for the cosine
 and a similar identity holds for sin(as).
Derivatives of the delta function
The derivative of the Dirac delta function (also called a doublet) is the distribution δ' defined by
for every test function . From this it follows that
The nth derivative δ^{(n)} is given by
The derivatives of the Dirac delta are important because they appear in the Fourier transforms of polynomials.
Representations of the delta function
The delta function can be viewed as the limit of a sequence of functions
where δ_{a}(x) is sometimes called a nascent delta function. This limit is in the sense that
for all continuous f.
The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation (on groups more general than the real numbers, e.g. the unit circle). There the condition is made that the limiting sequence should be of positive functions.
Some nascent delta functions are:

Limit of a Normal distribution Limit of a Cauchy distribution Cauchy (see note below) Limit of a rectangular function rectangular function (see note below) Derivative of the sigmoid (or FermiDirac) function Limit of the Airy function Limit of a Bessel function
Note: If δ(a, x) is a nascent delta function which is a probability distribution over the whole real line (i.e. is always nonnegative between ∞ and +∞) then another nascent delta function δ_{φ}(a, x) can be built from its characteristic function as follows:
where
is the characteristic function of the nascent delta function δ(a, x). This result is related to the localization property of the continuous Fourier transform.
The Dirac comb
 Main article: Dirac comb
A socalled uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis.
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