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Generalized function whose value is zero everywhere except at zero
the Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the
Dirac_delta_function
Periodic distribution ("function") of "point-mass" Dirac delta sampling
{\displaystyle k} . The Dirac delta function δ {\displaystyle \delta } and the Dirac comb are tempered distributions. The graph of the function resembles a comb
Dirac_comb
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
continuous-time systems the Dirac delta function is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as:
Kronecker_delta
Indicator function of positive numbers
integral of the Dirac delta function. This is sometimes written as: H ( x ) := ∫ − ∞ x δ ( s ) d s , {\displaystyle H(x):=\int _{-\infty }^{x}\delta (s)\,ds,}
Heaviside_step_function
Measure that is 1 if and only if a specified element is in the set
of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. A Dirac measure is a measure δx on a set
Dirac_measure
Model of an energy potential in quantum mechanics
quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it
Delta_potential
Output of a dynamic system when given a brief input
function contains all frequencies (see the Fourier transform of the Dirac delta function, showing infinite frequency bandwidth that the Dirac delta function
Impulse_response
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
{\displaystyle \delta (t)} is δ ( f ) = 1 , {\displaystyle \delta (f)=1,} means that the frequency spectrum of the Dirac delta function is infinitely broad
Rectangular_function
Function returning minus 1, zero or plus 1
{sgn}(x)F(x)-\int {2\delta (x)F(x){\text{d}}x}\,,} where δ ( x ) {\textstyle \delta (x)} is the Dirac delta function. Integrating, the following
Sign_function
Fundamental object of geometry
as points with non-zero charge). The Dirac delta function, or δ function, is (informally) a generalized function on the real number line that is zero
Point_(geometry)
Method of solution to differential equations
function G {\displaystyle G} is the solution of the equation L G = δ , {\displaystyle LG=\delta ,} where δ {\displaystyle \delta } is Dirac's delta function;
Green's_function
Topics referred to by the same term
A Dirac delta function or simply delta function is a generalized function on the real number line denoted by δ that is zero everywhere except at zero
Delta function (disambiguation)
Delta_function_(disambiguation)
Limit of sequence of smooth functions
on the indicator function of some domain D. It is a generalisation of the derivative (or "prime function") of the Dirac delta function to higher dimensions;
Laplacian_of_the_indicator
British physicist (1902–1984)
career, Dirac made numerous important contributions to mathematical subjects, including the Dirac delta function, Dirac algebra and the Dirac operator
Paul_Dirac
Fourth letter in the Greek alphabet
The Kronecker delta in mathematics. The central difference for a function. The degree of a vertex in graph theory. The Dirac delta function in mathematics
Delta_(letter)
arguments. The integral of the Dirac delta function. Sawtooth wave Square wave Triangle wave Rectangular function Floor function: Largest integer less than
List of mathematical functions
List_of_mathematical_functions
Mathematical function characterizing set membership
step function is equal to the Dirac delta function, i.e. d H ( x ) d x = δ ( x ) {\displaystyle {\frac {\mathrm {d} H(x)}{\mathrm {d} x}}=\delta (x)}
Indicator_function
Mathematical description of quantum state
potentials that are not functions but are distributions, such as the Dirac delta function. It is easy to visualize a sequence of functions meeting the requirement
Wave_function
Multivalued function in mathematics
provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges—a fundamental problem in physics. Prompted
Lambert_W_function
French mathematician (1915–2002)
of distributions or generalized functions, giving a well-defined meaning to objects such as the Dirac delta function. For several years he taught at the
Laurent_Schwartz
Parametrization used for loop integrals
electrodynamics. Hung Cheng and T.T. Wu proved in 1987 that the sum in the Dirac delta function can be reduced to a subset of Feynman parameters. This result is
Feynman_parametrization
Objects extending the notion of functions
1920s and 1930s further basic steps were taken. The Dirac delta function was boldly defined by Paul Dirac (an aspect of his scientific formalism); this was
Generalized_function
Integral transform useful in probability theory, physics, and engineering
special case is where μ is a probability measure, for example, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often
Laplace_transform
Probability distribution
at a specific point (that is its probability distribution is the Dirac delta function), then after time t its location is described by a normal distribution
Normal_distribution
Objects that generalize functions
distributions, such as the Dirac delta function. The practical use of distributions can be traced back to the use of Green functions in the 1830s to solve
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Partial differential equations
three-dimensional space, and δ {\displaystyle \delta } is the Dirac delta function. The algebraic expression of the Green's function for the three-variable Laplace operator
Green's function for the three-variable Laplace equation
Green's_function_for_the_three-variable_Laplace_equation
Special mathematical functions defined on the surface of a sphere
indices and the Dirac delta function. For the spherical harmonics, the Dirac delta is the tensor product of two Dirac delta functions, one for the azimuthal
Spherical_harmonics
Characteristic time in a system
the step response to a step input, or the impulse response to a Dirac delta function input. In the frequency domain (for example, looking at the Fourier
Time_constant
Probability distribution
This function is also known as a Lorentzian function, and an example of a nascent delta function, and therefore approaches a Dirac delta function in the
Cauchy_distribution
Description of continuous random distribution
the probability density function of X {\displaystyle X} and δ ( ⋅ ) {\displaystyle \delta (\cdot )} be the Dirac delta function. It is possible to use
Probability_density_function
Probability distribution
distribution becomes a one-point degenerate distribution with a Dirac delta function spike at the right end, x = 1, with probability 1, and zero probability
Beta_distribution
Mathematical concept
convolution quotients allows easy algebraic representation of the Dirac delta function, integral operator, and differential operator without having to deal
Convolution_quotient
Topics referred to by the same term
distribution of a function Difference operator (Δ) Dirac delta function (δ function) Increment operator (∆) Kronecker delta ( δ i j {\displaystyle \delta _{ij}}
Delta
functions. Symmetric function: value is independent of the order of its arguments Generalized function: a wide generalization of Dirac delta function
List_of_types_of_functions
Function in quantum field theory showing probability amplitudes of moving particles
t')=\delta (x-x')\delta (t-t'),} where H denotes the Hamiltonian, δ(x) denotes the Dirac delta-function and Θ(t) is the Heaviside step function. The kernel
Propagator
Operator in quantum mechanics
{\displaystyle x} is the Dirac delta (function) distribution centered at the position x {\displaystyle x} , often denoted by δ x {\displaystyle \delta _{x}} . In quantum
Position_operator
Hypothetical particle with one magnetic pole
magnetic field is proportional to the Dirac delta function at the origin. We must define one set of functions for the vector potential on the "northern
Magnetic_monopole
Extremely small quantity in calculus; thing so small that there is no way to measure it
continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstract versions of
Infinitesimal
Partial differential equation describing the evolution of temperature in a region
{R} \times (0,\infty )\\u(x,0)=\delta (x)&\end{cases}}} where δ {\displaystyle \delta } is the Dirac delta function. The fundamental solution to this
Heat_equation
First-order differential linear operator on spinor bundle, whose square is the Laplacian
= Δ + m 2 {\displaystyle D^{2}=\Delta +m^{2}} (after setting ℏ = c = 1. {\displaystyle \hbar =c=1.} ) Another Dirac operator arises in Clifford analysis
Dirac_operator
Family of solutions to related differential equations
approaches zero, the right-hand side approaches δ(x − 1), where δ is the Dirac delta function. This admits the limit (in the distributional sense): ∫ 0 ∞ k J α
Bessel_function
Mathematical transform that expresses a function of time as a function of frequency
relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically
Fourier_transform
Mathematical function for the probability a given outcome occurs in an experiment
distributions can be represented with the Dirac delta function as a generalized probability density function f {\displaystyle f} , where f ( x ) = ∑ ω
Probability_distribution
Reactor simulation model
the plug is a function of its position in the reactor. In the ideal PFR, the residence time distribution is therefore a Dirac delta function with a value
Plug_flow_reactor_model
Mathematical function having a characteristic "bell"-shaped curve
functions with decreasing variance that approach the Dirac delta distribution. Indeed, the Dirac delta can roughly be thought of as a bell curve with variance
Bell-shaped_function
integral Dirac delta function Dirac comb Dirac measure Dirac operator Dirac algebra 5997 Dirac, an asteroid The various Dirac Medals Dirac (software) DiRAC supercomputing
List of things named after Paul Dirac
List_of_things_named_after_Paul_Dirac
Characteristic of an optical system
function diverges at the origin x = y = z = 0. The function values along the z-axis of the 3D optical transfer function correspond to the Dirac delta
Optical_transfer_function
Electric circuit composed of resistors and capacitors
h_{R}(t)=\delta (t)-{\frac {1}{RC}}e^{-{\frac {t}{RC}}}u(t)=\delta (t)-{\frac {1}{\tau }}e^{-{\frac {t}{\tau }}}u(t)\,,} where δ(t) is the Dirac delta function
RC_circuit
Inputs for which a function's value is non-zero
density function. It is possible also to talk about the support of a distribution, such as the Dirac delta function δ ( x ) {\displaystyle \delta (x)} on
Support_(mathematics)
Paradigmatic model
T} is the kicking period and δ {\displaystyle \textstyle \delta } is the Dirac delta function. The equations of motion of the kicked rotator write d θ
Kicked_rotator
Concept in the solution of linear partial differential equations
Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta function δ(x)
Fundamental_solution
Partial differential equations with random force terms and coefficients
as ∂ t u = Δ u + ξ , {\displaystyle \partial _{t}u=\Delta u+\xi \;,} where Δ {\displaystyle \Delta } is the Laplacian and ξ {\displaystyle \xi } denotes
Stochastic partial differential equation
Stochastic_partial_differential_equation
Condition to avoid intersymbol interference
{\displaystyle n} . We multiply such a h(t) by a sum of Dirac delta function (impulses) δ ( t ) {\displaystyle \delta (t)} separated by intervals Ts This is equivalent
Nyquist_ISI_criterion
Sum of a function's values every _P_ offsets
summation of a Dirac delta function is the Dirac comb. Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb. If
Periodic_summation
Idealised model of a particle in physics
such as mass or charge, it is often represented mathematically by a Dirac delta function. In classical mechanics there is usually no concept of rotation of
Point_particle
Fundamental physical law of electromagnetism
{\mathbf {r} }{|\mathbf {r} |^{3}}}\right)=4\pi \delta (\mathbf {r} )} where δ(r) is the Dirac delta function, the result is ∇ ⋅ E ( r ) = 1 ε 0 ∫ ρ ( s )
Coulomb's_law
Quick, temporary change in amplitude of electrical signals
A Dirac pulse has the shape of the Dirac delta function. It has the properties of infinite amplitude and its integral is the Heaviside step function. Equivalently
Pulse_(signal_processing)
1). The Dirac delta function, although not strictly a probability distribution, is a limiting form of many continuous probability functions. It represents
List of probability distributions
List_of_probability_distributions
Distributions on spaces of differential forms
submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets
Current_(mathematics)
Complex analysis theorem
}}=\mp i\pi \delta (x)+{\mathcal {P}}{{\Big (}{\frac {1}{x}}{\Big )}}.} where δ ( x ) {\displaystyle \delta (x)} is the Dirac delta function where P {\displaystyle
Sokhotski–Plemelj_theorem
Transition rate formula
\varepsilon |\varepsilon '\rangle =\delta (\varepsilon -\varepsilon ')} where δ {\displaystyle \delta } is the Dirac delta function, and effectively a factor of
Fermi's_golden_rule
Relativistic quantum mechanical wave equation
wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers (known as Dirac spinors)
Dirac_equation
Theoretical framework in physics
},{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=0,} where δ is the Dirac delta function. The vacuum state | 0 ⟩ {\displaystyle |0\rangle } is defined by
Quantum_field_theory
following functions and variables are used in the table below: δ represents the Dirac delta function. u(t) represents the Heaviside step function. Literature
List_of_Laplace_transforms
Reconstruction of a filtered signal
estimated wavelet to a Dirac delta function (i.e., a spike). The result may be seen as a series of scaled, shifted delta functions (although this is not
Deconvolution
Differential equation for the description of waves or standing wave
s(t,x)=\delta ^{D+1}(t,x)} where δ {\displaystyle \delta } is the Dirac delta function. The solution to this case is called the Green's function G {\displaystyle
Wave_equation
Relative importance of certain frequencies in a composite signal
=2\pi f(\omega )\delta (\omega -\omega '),} where δ ( ω − ω ′ ) {\displaystyle \delta (\omega -\omega ')} is the Dirac delta function. Such formal statements
Spectral_density
Result about when a matrix can be diagonalized
f(t)=\delta (t-t_{0})} , where δ {\displaystyle \delta } is the Dirac delta function, is an eigenvector when construed in an appropriate sense. The Dirac delta
Spectral_theorem
this function for different values of n reveals that as n goes to infinity, L n ( t ) {\displaystyle L_{n}(t)} approaches the Dirac delta function as a
Landau_kernel
Taylor series expansion in probability theory
}{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x-x_{0})\mu _{n}(t|x_{0},t_{0})} Now we need to integrate away the Dirac delta function. Fixing a small τ > 0 {\displaystyle
Kramers–Moyal_expansion
Mathematics award
theory of distributions, a new notion of generalized function motivated by the Dirac delta-function of theoretical physics." Atle Selberg Institute for
Fields_Medal
Type of signal in signal processing
the power spectral density and δ {\displaystyle \delta } is the Dirac delta function, an unbounded measure which correctly reflects the infinite variance
White_noise
Distribution of distances between pairs of particles in a given volume
is therefore a set of Dirac delta functions of the form: g ( r ) = ∑ i δ ( r − i b ) {\displaystyle g(r)=\sum \limits _{i}\delta (r-ib)} . Finally, it
Pair_distribution_function
Integral expressing the amount of overlap of one function as it is shifted over another
convolution with a translated Dirac delta function τxf = f ∗ τx δ. So translation invariance of the convolution of Schwartz functions is a consequence of the
Convolution
Numerical parameter in probability theory
concentrated on a single point, the degenerate distribution defined by the Dirac delta function). In the case of multivariate Dirichlet distributions, there is some
Concentration_parameter
Eigenvalue problem for the Laplace operator
the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with f equaling the Dirac delta function, so G satisfies
Helmholtz_equation
Foundational principle in quantum physics
wave function vanishes at both infinities and | e − i p χ / ℏ | = 1 {\displaystyle |e^{-ip\chi /\hbar }|=1} , and then use the Dirac delta function which
Uncertainty_principle
Rate of change of acceleration with time
be modeled using a Dirac delta function in jerk, scaled to the height of the jump. Integrating jerk over time across the Dirac delta yields the jump-discontinuity
Jerk_(physics)
Concept in calculus of variations
{\delta f^{-1}(x)}{\delta f(y)}}=-{\frac {\delta \left(f^{-1}(x)-y\right)}{f'\left(f^{-1}(x)\right)}}} In physics, it is common to use the Dirac delta function
Functional_derivative
Spontaneous breakdown of an unstable subatomic particle into other particles
)^{3}E_{i}}}} where δ 4 {\displaystyle \delta ^{4}\,} is a four-dimensional Dirac delta function, p → i {\displaystyle {\vec {p}}_{i}\,} is the (three-)momentum of
Particle_decay
Generalized version of classical Green's function
function of two discrete variable m and n. Similar to the case of Dirac delta function for continuous variables, it is defined to be 1 if m = n and 0 otherwise
Multiscale_Green's_function
Description of a quantum-mechanical system
Likewise a position eigenstate would be a Dirac delta distribution, not square-integrable and technically not a function at all. Consequently, neither can belong
Schrödinger_equation
General function in mathematics
In mathematics, the unit doublet is a generalized function, the derivative of the Dirac delta function. It can be used to differentiate signals in electrical
Unit_doublet
Decomposition of periodic functions
{\displaystyle {\mathcal {F}}\{e^{i2\pi {\tfrac {n}{P}}x}\}} is a Dirac delta function, which is an example of a distribution. "Fourier". Dictionary.com
Fourier_series
Statistical description for the behavior of fermions
Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles
Fermi–Dirac_statistics
Variant of the Laplace integral transform
analysis of certain functions, particularly the unit step function and Dirac delta function, whose transforms become simple constants. The transform has
Laplace–Carson_transform
Conversion of continuous functions into discrete counterparts
tempered distribution (e.g. a Dirac delta function δ {\displaystyle \delta } or any other compactly supported function), α {\displaystyle \alpha } is
Discretization
Distribution of variables which satisfies a stability property under linear combinations
bound corresponding to the normal distribution, and approaches the Dirac delta function in the limit as α → 0 {\displaystyle \alpha \rightarrow 0} . The
Stable_distribution
Frequency domain representation of random fluctuations in the phase of a waveform
frequency domain, this would be represented as a single pair of Dirac delta functions (positive and negative conjugates) at the oscillator's frequency;
Phase_noise
Stochastic process generalizing Brownian motion
\\&\operatorname {E} [\xi (t)\xi (s)]=\delta (t-s),\end{aligned}}} with the Dirac delta "function" δ {\displaystyle \delta } . However, as shown below, the
Wiener_process
Fourier transform of the probability density function
_{X}^{(n)}(0),\!} This can be formally written using the derivatives of the Dirac delta function: f X ( x ) = ∑ n = 0 ∞ ( − 1 ) n n ! δ ( n ) ( x ) E [ X n ] {\displaystyle
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Foundational law of electromagnetism relating electric field and charge distributions
{\mathbf {r} }{|\mathbf {r} |^{3}}}\right)=4\pi \delta (\mathbf {r} )} where δ(r) is the Dirac delta function, the result is ∇ ⋅ E ( r ) = 1 ε 0 ∫ ρ ( s )
Gauss's_law
Physics models of a 1D gas of bosons
statistics. More specifically, it describes a one dimensional Bose gas with Dirac delta interactions. It is named after Elliott H. Lieb and Werner Liniger [de]
Lieb–Liniger_model
Mathematical function
diffusion. Specifically, if the mass-density at time t=0 is given by a Dirac delta, which essentially means that the mass is initially concentrated in a
Gaussian_function
Second-order partial differential equation
′ , z − z ′ ) , {\displaystyle \Delta u=u_{xx}+u_{yy}+u_{zz}=-\delta (x-x',y-y',z-z'),} where the Dirac delta function δ denotes a unit source concentrated
Laplace's_equation
Topics referred to by the same term
spin of 1⁄2 Unit doublet, in mathematics, the derivative of the Dirac delta function Doublet (horse), (b. 1963 - †.1974), a thoroughbred gelding Doublet
Doublet
Random motion of particles suspended in a fluid
squared displacement: E [ ( Δ x ) 2 ] {\textstyle \mathbb {E} {\left[(\Delta x)^{2}\right]}} . However, when he relates it to a particle of mass m moving
Brownian_motion
Function of propagation delay and Doppler frequency
ambiguity function of interest is a 2-dimensional Dirac delta function or "thumbtack" function; that is, a function which is infinite at (0,0) and zero elsewhere
Ambiguity_function
For an infinite crystal, the diffracted pattern is concentrated in Dirac delta function like Bragg peaks. Presence of crystalline surfaces results in additional
X-ray_crystal_truncation_rod
Mathematical function of a linear operator
product of the eigenfunctions equal to either a Kronecker delta or a Dirac delta function, respectively. For many Hermitian operators, notably Sturm–Liouville
Eigenfunction
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DIRAC DELTA-FUNCTION
DIRAC DELTA-FUNCTION
DIRAC DELTA-FUNCTION
DIRAC DELTA-FUNCTION
DIRAC DELTA-FUNCTION
DIRAC DELTA-FUNCTION
DIRAC DELTA-FUNCTION
DIRAC DELTA-FUNCTION
DIRAC DELTA-FUNCTION
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