# Dirac delta function

The **Dirac delta function**, written \( \delta(x) \), is a mathematical object that is zero for every \( x \neq 0 \) yet integrates to one over any interval containing the origin. No ordinary function has these properties, so the delta is rigorously defined either as a distribution (a generalized function) or as a measure, and it is meaningful only through its action under an integral sign.<sup>[1](https://books.physics.oregonstate.edu/LinAlg/deltaintro.html)</sup> It was introduced informally by the physicist [Paul Dirac](https://www.edgepedia.org/paul-dirac) and made rigorous by the theory of distributions developed in the mid twentieth century.<sup>[2](https://www.mat.univie.ac.at/~stein/teaching/courses/WS1718/dp.php)</sup>

| Key fact | Detail |
|---|---|
| Informal picture | \( \delta(x) = 0 \) for \( x \neq 0 \), with total integral 1 over any interval containing 0; no classical function satisfies both conditions<sup>[1](https://books.physics.oregonstate.edu/LinAlg/deltaintro.html)</sup> |
| Defining action (sifting property) | \( \int_{-\infty}^{\infty} \delta(x-a)\,\varphi(x)\,dx = \varphi(a) \) for suitable test functions \( \varphi \)<sup>[3](https://dlmf.nist.gov/1.17)</sup> |
| Relation to Heaviside step | The delta is the derivative of the Heaviside function \( h \) in the sense of distributions<sup>[4](https://encyclopediaofmath.org/wiki/Delta-function)</sup> |
| Rigorous status | A singular distribution: a linear functional on test functions with \( \langle \delta, \varphi \rangle = \varphi(0) \)<sup>[5](https://dlmf.nist.gov/1.16)</sup> |
| Approximation | Families of narrow unit-area functions converge to the delta; one delta sequence is \( \delta_n(x-a) = \sqrt{n/\pi}\, e^{-n(x-a)^2} \)<sup>[3](https://dlmf.nist.gov/1.17)</sup> |
| Origin | Used by Dirac in quantum mechanics (The Principles of Quantum Mechanics, 1930); made rigorous by Laurent Schwartz's distribution theory around 1945<sup>[6](https://mathshistory.st-andrews.ac.uk/Obituaries/Schwartz/)</sup><sup> • </sup><sup>[2](https://www.mat.univie.ac.at/~stein/teaching/courses/WS1718/dp.php)</sup> |
| Main uses | Point sources in differential equations (Green's functions), impulse response in signal processing and control, position eigenstates in quantum mechanics<sup>[7](https://www.maths.dur.ac.uk/users/christopher.prior/quarto-for-chris/notes/_book/greens.html)</sup><sup> • </sup><sup>[8](https://courses.physics.illinois.edu/ece486/fa2026/documentation/handbook/lec03.html)</sup> |

## Definition and the sifting property

The most useful definition is operational. For a point located at \( x = a \), the delta is defined by what it does when integrated against a continuous function:

\[ \int_{-\infty}^{\infty} \delta(x-a)\, f(x)\, dx = f(a). \]

This is called the sifting property: under the integral, the delta picks out the value of \( f \) at the point where the delta is located.<sup>[7](https://www.maths.dur.ac.uk/users/christopher.prior/quarto-for-chris/notes/_book/greens.html)</sup><sup> • </sup><sup>[9](https://www.maths.ox.ac.uk/system/files/attachments/ASO_IT_lectures20.pdf)</sup> Taking \( f(x) = 1 \) gives the unit-area condition \( \int_{-\infty}^{\infty} \delta(x-a)\,dx = 1 \). The shorthand \( \delta(x-a) = 0 \) for \( x \neq a \) should be read as a statement about where the delta is concentrated, not as a pointwise definition.<sup>[7](https://www.maths.dur.ac.uk/users/christopher.prior/quarto-for-chris/notes/_book/greens.html)</sup>

The [National Institute of Standards and Technology](https://www.edgepedia.org/national-institute-of-standards-and-technology)'s Digital Library of Mathematical Functions presents the delta as a symbolic function with exactly these two properties, \( \delta(x) = 0 \) for \( x \neq 0 \) and the sifting integral above, noting that the integral is interpreted as a generalized limit of integrals against ordinary functions.<sup>[3](https://dlmf.nist.gov/1.17)</sup>

## Why it is not a function

A function that is nonzero at only one point integrates to zero, if it is integrable at all; a function that is infinite at one point is not an ordinary integrable function. So no classical integrable function behaves as the delta does, and the delta must be a different kind of object, defined through its action under an integral.<sup>[7](https://www.maths.dur.ac.uk/users/christopher.prior/quarto-for-chris/notes/_book/greens.html)</sup> In the language of distribution theory, a distribution is a linear functional on a space of test functions, and the delta is the functional \( \delta : \varphi \mapsto \varphi(0) \). It is a singular distribution, meaning it does not arise from integrating against a locally integrable function.<sup>[2](https://www.mat.univie.ac.at/~stein/teaching/courses/WS1718/dp.php)</sup><sup> • </sup><sup>[5](https://dlmf.nist.gov/1.16)</sup>

**Delta sequences.** The delta can be approached by ordinary functions. A sequence \( \delta_n \) is a delta sequence if, for every suitable test function \( \varphi \),

\[ \lim_{n \to \infty} \int_{-\infty}^{\infty} \delta_n(x-a)\,\varphi(x)\,dx = \varphi(a). \]

A standard example is a family of normalized Gaussians whose widths shrink as \( n \) grows,

\[ \delta_n(x-a) = \sqrt{\frac{n}{\pi}}\, e^{-n(x-a)^2}. \]

Each member has unit area, and the limit of the integrals is \( \varphi(a) \).<sup>[3](https://dlmf.nist.gov/1.17)</sup> The same idea appears in engineering treatments as a limit of rectangular pulses of width \( \varepsilon \) and height \( 1/\varepsilon \): as \( \varepsilon \to 0 \) the pulse grows taller without bound while its area stays 1.<sup>[8](https://courses.physics.illinois.edu/ece486/fa2026/documentation/handbook/lec03.html)</sup>

## Calculus with the delta

Distribution theory extends differentiation to objects that have no classical derivative. The derivative of a distribution \( \Lambda \) is defined by \( \langle \Lambda', \varphi \rangle = -\langle \Lambda, \varphi' \rangle \), a formula modeled on integration by parts. Under this definition the delta is the derivative of the [Heaviside step function](https://www.edgepedia.org/heaviside-step-function) \( H \), which is 1 for \( x > 0 \) and 0 for \( x \leq 0 \): \( \mathbf{D}H = \delta \).<sup>[5](https://dlmf.nist.gov/1.16)</sup><sup> • </sup><sup>[4](https://encyclopediaofmath.org/wiki/Delta-function)</sup> Higher derivatives of the delta are likewise well defined, with \( \langle \delta_{x_0}^{(n)}, \varphi \rangle = (-1)^n \varphi^{(n)}(x_0) \).<sup>[5](https://dlmf.nist.gov/1.16)</sup>

The [Fourier transform](https://www.edgepedia.org/fourier-transform) gives another representation. With the convention used in MIT's quantum theory notes,

\[ \delta(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}\, dk, \]

so the delta can be defined as the Fourier transform of the identity, that is, of the constant function 1.<sup>[10](https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/fd01a08a88e3e030eeb3a999b5339170_MIT8_321F17_Rec2.pdf)</sup> The DLMF catalogs many further integral and series representations, including cosine and sine integral forms and expansions in orthogonal polynomials such as

\[ \delta(x-a) = \sum_{k=0}^{\infty} \left(k + \tfrac{1}{2}\right) P_k(x)\, P_k(a) \]

in [Legendre polynomials](https://www.edgepedia.org/legendre-polynomials), together with analogous formulas in Laguerre, Hermite, and spherical harmonics.<sup>[3](https://dlmf.nist.gov/1.17)</sup>

In \( d \) dimensions the delta factors coordinatewise, \( \delta(\mathbf{x}) = \delta(x_1)\,\delta(x_2)\cdots\delta(x_d) \).<sup>[11](https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/5bb33034c75768ba03e140bb7d848321_MIT18_306f09_assn02_ProblemList20080319.pdf)</sup>

## History

The delta entered mathematics through physics. Ideas of impulsive and concentrated sources circulated long before a rigorous theory: Kirchhoff in 1882, Heaviside in 1898, and Dirac in 1926 all used generalized-function notions in a vague sense. Dirac employed his "delta functions" as formal tools in quantum mechanics, including in *[The Principles of Quantum Mechanics](https://www.edgepedia.org/the-principles-of-quantum-mechanics)* (1930); as Dirac himself knew, in strict mathematical terms they made no sense, but they worked.<sup>[2](https://www.mat.univie.ac.at/~stein/teaching/courses/WS1718/dp.php)</sup><sup> • </sup><sup>[6](https://mathshistory.st-andrews.ac.uk/Obituaries/Schwartz/)</sup>

The rigorous foundation came from the theory of distributions, developed in the mid twentieth century mainly by [Laurent Schwartz](https://www.edgepedia.org/laurent-schwartz) and [Sergei Sobolev](https://www.edgepedia.org/sergei-sobolev). Schwartz's functional-analytic formulation, around 1945, defines a generalized function as a linear functional on a space of test functions and thereby gives the delta a precise meaning; within a few years distributions became a central topic of functional analysis and an indispensable tool in partial differential equations and theoretical physics.<sup>[2](https://www.mat.univie.ac.at/~stein/teaching/courses/WS1718/dp.php)</sup>

## Applications

**Green's functions.** A Green's function describes the response of a linear differential operator to a point source, and the point source is represented by a delta. For a boundary-value problem \( Ly = f \), one solves \( Lg = \delta(x - \xi) \) for the [Green's function](https://www.edgepedia.org/greens-function) \( g(x,\xi) \); the solution for an arbitrary source is then the convolution \( y(x) = \int g(x,\xi) f(\xi)\, d\xi \), because the delta's sifting property reconstructs \( f \) from a continuum of point sources.<sup>[7](https://www.maths.dur.ac.uk/users/christopher.prior/quarto-for-chris/notes/_book/greens.html)</sup><sup> • </sup><sup>[12](https://physicstoday.aip.org/features/the-green-of-green-functions)</sup> For the Laplacian, the fundamental solution \( \Gamma \) satisfies \( -\Delta\Gamma = \delta \) in the sense of distributions and represents the potential of a point charge at the origin.<sup>[13](https://math.ucdavis.edu/~hunter/pdes/pde_notes.pdf)</sup> For the wave equation in \( d \) dimensions, the Green's functions carry the delta as their initial displacement or initial velocity.<sup>[11](https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/5bb33034c75768ba03e140bb7d848321_MIT18_306f09_assn02_ProblemList20080319.pdf)</sup>

**Signals and control.** In linear time-invariant systems, the response to a unit impulse \( \delta(t) \) is called the impulse response \( h \). Because any reasonable input can be written as an integral of shifted deltas via the sifting property, the output of such a system with zero initial conditions is the convolution of the input with \( h \), and the [Laplace transform](https://www.edgepedia.org/laplace-transform) of \( h \) is the system's transfer function.<sup>[8](https://courses.physics.illinois.edu/ece486/fa2026/documentation/handbook/lec03.html)</sup>

**Quantum mechanics.** The eigenvectors of the position operator take the form of delta functions, and delta normalization such as \( \langle x \mid x' \rangle = \delta(x - x') \) underlies the continuous-spectrum formalism of quantum theory.<sup>[14](https://arxiv.org/abs/1808.05165)</sup>

## Limits of the formalism

Multiplication is the sharp edge of distribution theory. Multiplying a distribution by a smooth function is always well defined in Schwartz's theory, but the Schwartz impossibility result shows that there is no associative product of two distributions that extends this well-defined product; expressions such as \( \delta(x - x')\,\delta(x - x'') \), which share a common variable, are not well defined.<sup>[15](https://plato.stanford.edu/entries/qt-nvd/)</sup><sup> • </sup><sup>[16](https://encyclopediaofmath.org/wiki/Multiplication_of_distributions)</sup> This limitation is one motivation for later theories of generalized functions, such as Colombeau's algebra, in which a general multiplication of distributions is defined.<sup>[17](https://arxiv.org/abs/math/0603351)</sup>

## References

1. [The Dirac Delta Function, Oregon State University Physics](https://books.physics.oregonstate.edu/LinAlg/deltaintro.html)
2. [Theory of Distributions 2017, R. Steinbauer and M. Kunzinger, University of Vienna](https://www.mat.univie.ac.at/~stein/teaching/courses/WS1718/dp.php)
3. [DLMF §1.17: Integral and Series Representations of the Dirac Delta, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/1.17)
4. [Delta-function, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Delta-function)
5. [DLMF §1.16: Distributions, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/1.16)
6. [Laurent Schwartz, Times obituary, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Obituaries/Schwartz/)
7. [Green's method, MATH 2811 Mathematical Methods II, C. Prior, Durham University](https://www.maths.dur.ac.uk/users/christopher.prior/quarto-for-chris/notes/_book/greens.html)
8. [ECE 486 Control Systems, Lecture 3, University of Illinois](https://courses.physics.illinois.edu/ece486/fa2026/documentation/handbook/lec03.html)
9. [Distribution theory lecture notes, Mathematical Institute, University of Oxford](https://www.maths.ox.ac.uk/system/files/attachments/ASO_IT_lectures20.pdf)
10. [Quantum Theory I, Recitation 2 Notes, MIT OpenCourseWare (8.321, Fall 2017)](https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/fd01a08a88e3e030eeb3a999b5339170_MIT8_321F17_Rec2.pdf)
11. [Point Sources and Green Functions, 18.306 Problem List, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-306-advanced-partial-differential-equations-with-applications-fall-2009/5bb33034c75768ba03e140bb7d848321_MIT18_306f09_assn02_ProblemList20080319.pdf)
12. [The Green of Green Functions, Physics Today](https://physicstoday.aip.org/features/the-green-of-green-functions)
13. [PDE notes, J. Hunter, University of California, Davis](https://math.ucdavis.edu/~hunter/pdes/pde_notes.pdf)
14. [Energy transfer and position measurement in quantum mechanics, arXiv:1808.05165](https://arxiv.org/abs/1808.05165)
15. [Quantum Theory and Mathematical Rigor, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/qt-nvd/)
16. [Multiplication of distributions, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Multiplication_of_distributions)
17. [Distributions with dynamic test functions and multiplication by discontinuous functions, arXiv:math/0603351](https://arxiv.org/abs/math/0603351)

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