Dirac delta function

E199880
distribution generalized function idealized function mathematical concept tempered distribution

The Dirac delta function is a mathematical construct used in physics and engineering to model an idealized point mass or point charge, being zero everywhere except at a single point where it is infinitely large yet integrates to one.

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All labels observed (6)

Label Occurrences
Dirac delta function canonical 4
Dirac delta distribution 2
Dirac comb 1

Statements (60)

Predicate Object
instanceOf distribution ⓘ
generalized function ⓘ
idealized function ⓘ
mathematical concept ⓘ
tempered distribution ⓘ
actsOn test functions ⓘ
alsoKnownAs Dirac delta function ⓘ
surface form: Dirac delta

Dirac delta function ⓘ
surface form: delta function

impulse function ⓘ
appearsIn Green's function methods ⓘ
Maxwell's equations with point charges ⓘ
Poisson equation ⓘ
surface form: Poisson's equation

Schrödinger equation with point interactions ⓘ
approximationSequence narrow Gaussian functions with unit area ⓘ
rectangular pulses with shrinking width and fixed area ⓘ
sinc-based kernels in limit ⓘ
characterizedBy defined as a linear functional on test functions ⓘ
integral over entire real line equals one ⓘ
not a function in the classical sense ⓘ
zero everywhere except at a single point ⓘ
codomain space of distributions ⓘ
convolutionIdentity f * δ = f for suitable functions f ⓘ
definingProperty ∫_{-∞}^{∞} δ(x) φ(x) dx = φ(0) for test functions φ ⓘ
∫_{-∞}^{∞} δ(x-a) φ(x) dx = φ(a) ⓘ
derivative distribution δ′ (delta prime) ⓘ
distributionOrder 0 ⓘ
domain distribution theory ⓘ
functional analysis ⓘ
real analysis ⓘ
evenFunction true ⓘ
FourierTransform constant function 1 (in suitable normalization) ⓘ
generalizationOf Kronecker delta ⓘ
surface form: Kronecker delta (discrete case)
introducedInContext quantum mechanics ⓘ
inverseFourierTransform constant function 1 (in suitable normalization) ⓘ
LaplaceTransform 1 ⓘ
linearity linear functional ⓘ
mathematicalFramework theory of distributions by Laurent Schwartz ⓘ
models idealized point charge ⓘ
idealized point mass ⓘ
instantaneous impulse ⓘ
namedAfter Paul Dirac ⓘ
relatedConcept Heaviside step function ⓘ
unit impulse in discrete time ⓘ
scalingProperty δ(ax) = δ(x)/|a| for nonzero a ⓘ
support single point ⓘ
{0} ⓘ
supportType compact support ⓘ
symbol δ ⓘ
testFunctionSpace space of smooth functions with compact support ⓘ
translationProperty δ(x-a) is delta centered at a ⓘ
usedIn classical mechanics ⓘ
control theory ⓘ
electrical engineering ⓘ
physics ⓘ
probability theory ⓘ
quantum mechanics ⓘ
signal processing ⓘ
systems theory ⓘ
usedToDefine Green's functions as responses to point sources ⓘ
impulse response of linear time-invariant systems ⓘ

Referenced by (10)

Full triples — surface form annotated when it differs from this entity's canonical label.

Paul Dirac → notableWork → Dirac delta function ⓘ
Kronecker delta → relatedTo → Dirac delta function ⓘ
Kronecker delta → isDiscreteAnalogOf → Dirac delta function ⓘ
Heaviside step function → derivativeInDistributionSense → Dirac delta function ⓘ
this entity surface form: Dirac delta distribution
Heaviside step function → relatedTo → Dirac delta function ⓘ
Dirac delta function → alsoKnownAs → Dirac delta function ⓘ
this entity surface form: delta function
Dirac delta function → alsoKnownAs → Dirac delta function ⓘ
this entity surface form: Dirac delta
Gelfand triples (rigged Hilbert spaces) → relatedTo → Dirac delta function ⓘ
subject surface form: Gelfand triple
this entity surface form: Dirac delta distribution
Poisson summation formula → involves → Dirac delta function ⓘ
this entity surface form: Dirac comb
measure theory → notableMeasure → Dirac delta function ⓘ
this entity surface form: Dirac measure