DiracDelta

The Dirac delta generalized function: 0 for any nonzero real argument, unevaluated at 0.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageDiracDelta!↗

DiracDelta(x)0 if x is a nonzero real number; unevaluated (stays symbolic) at x = 0 or for a symbolic argument.

DiracDelta(x1, x2, …)the multidimensional delta -- 0 as soon as any argument is a nonzero real, unevaluated otherwise.

Typed signatures

Type	Package	Applies
(real, real*) -> number	analytic	always

Domain: Elementary functions

Details

• DiracDelta has no ordinary value at 0 -- it is not a function in the usual sense, so DiracDelta(0) stays unevaluated rather than becoming Infinity or an error.
• D(HeavisideTheta(x), x) = DiracDelta(x) -- see HeavisideTheta.

Examples

Basic #

Unevaluated at 0

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In$\mathrm{DiracDelta}(0)$Out$\mathrm{DiracDelta}(0)$

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In$\mathrm{DiracDelta}(2)$Out$0$

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In$\mathrm{DiracDelta}(-3)$Out$0$

Zero away from the origin

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In$\mathrm{DiracDelta}(\frac{1}{2})$Out$0$

Scope #

Zero as soon as any argument is a nonzero real

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In$\mathrm{DiracDelta}(0, 1)$Out$0$

Zero at a negative point

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In$\mathrm{DiracDelta}(-1)$Out$0$

A multivariate delta is 0 when no argument is 0

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In$\mathrm{DiracDelta}(π, 8, \frac{1}{3})$Out$0$

The Fourier transform of a delta is constant

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In$\mathrm{FourierTransform}(\mathrm{DiracDelta}(x), x, t)$Out$\sqrt{\frac{1}{2π}}$

A shifted delta gives a unit-modulus phase factor $e^{iat}$

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In$\mathrm{FourierTransform}(\mathrm{DiracDelta}(x-a), x, t)$Out$\frac{\sqrt{2}\exp(i at)}{2\sqrt{π}}$

Applications #

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In$\int_{-∞}^{∞}\int_{-∞}^{∞}\!\mathrm{DiracDelta}(p^2+q^2-1)\, \mathrm{d}q \mathrm{d}p$Out$\int_{-∞}^{∞}\!\int_{-∞}^{∞}\!\mathrm{DiracDelta}(p^2+q^2-1)\, \mathrm{d}p\, \mathrm{d}q$

Properties #

Variable names do not matter; still constant

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In$\mathrm{FourierTransform}(\mathrm{DiracDelta}(y), y, x)$Out$\sqrt{\frac{1}{2π}}$

Zero at any nonzero real, however far out

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In$\mathrm{DiracDelta}(1)$Out$0$

Known value · Wolfram MathematicalFunctionData, DiracDelta

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In$\mathrm{DiracDelta}(0)\mathrm{DiracDelta}(\frac{3}{2})$Out$0$

Known value · Wolfram MathematicalFunctionData, DiracDelta, FunctionalEquations

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In$\mathrm{DiracDelta}(0)\mathrm{DiracDelta}(-\frac{1}{6})$Out$0$

Known value · Wolfram MathematicalFunctionData, DiracDelta, FunctionalEquations

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In$\mathrm{LaplaceTransform}(\mathrm{DiracDelta}(x), x, s)$Out$1$

Possible issues #

The argument is $e^{-e^{2e}} \approx 10^{-100}$, too close to $0$ for a double to tell: it stays unevaluated, as Wolfram leaves it

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In$\mathrm{DiracDelta}(-1-2e-e^{2}+\exp(-\exp(2e))+(1+e)^2)$Out$\mathrm{DiracDelta}(-1-2e-e^{2}+\exp(-\exp(2e))+(1+e)^2)$

About

Examples	16
Checked against	Wolfram Language 15 agree1 differ
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