# Dirac delta function The **Dirac delta function**, written δ(t) or δ(x) and also called the unit impulse, is a generalized function that is zero everywhere except at a single point, where it is not simply large but undefined as an ordinary value, and whose integral over the whole real line is exactly one. No ordinary function has that combination of properties; the delta is defined instead by what it does inside an integral against another function, picking out that function's value at one point. The primary microsim on this page builds the delta as a limit: the reader drags the width of an ordinary bump function toward zero and watches its height climb while the shaded area beneath it stays pinned at exactly 1.000, then watches a numerical integral converge on the exact value the delta is supposed to sift out. Despite not being a function in the everyday sense, the delta is one of the most heavily used objects in applied mathematics. It is the identity element for [[Convolution|convolution]], the model for an idealized instantaneous [[Impulse_response|impulse]] applied to a system, and the bridge between continuous and discrete signals in [[Sampling_(signal_processing)|sampling]]. ## Motivation and overview [[Physics|Physical]] and engineering reasoning constantly needs an idealization for something concentrated entirely at a single point or instant: a hammer blow delivered in an immeasurably short time but with a definite total impulse, a point charge with no spatial extent but a definite total charge, a single sample taken at one instant out of a continuous signal. An ordinary function cannot hold all three properties at once, zero width, a finite total, and one specific location, because a function with zero width at nonzero height integrates to zero, and a function with nonzero width has some genuine spatial extent. The Dirac delta resolves the contradiction by refusing to be an ordinary function at all: it is defined only by the one property that actually matters in every one of these use cases, that multiplying it against a smooth test function and integrating recovers that function's value at the delta's location. This is easiest to see as a limit. Take a family of ordinary bump functions, each of area exactly one and each narrower than the last; as the width shrinks toward zero, the family behaves more and more like a spike, and in the limit of that process, which is never actually reached by any single member of the family, the sifting behaviour that defines the delta emerges exactly. The common shorthand for writing δ(x) as "zero everywhere except at x = 0, where it is infinite" describes this limiting behaviour; it is not a literal description of any actual function. ## History Physicist Paul Dirac popularized the notation and the name in his 1930 textbook on quantum mechanics, using it to write the inner product of continuous position or momentum eigenstates in a compact form.[^dirac1930] Objects with a similar sifting behaviour had been used informally for a century or more before that, in the study of heat conduction, impulsive forces and Fourier's own integral theorem, but without a rigorous justification for treating something that was not a function as though it were one.[^cn_history] That justification arrived only in the late 1940s, when French mathematician Laurent Schwartz built a complete theory of generalized functions, or distributions, in which the delta is not an approximation or a shorthand but a well-defined object in its own right: a continuous linear functional on a space of smooth test functions. Schwartz's théorie des distributions earned him the Fields Medal at the 1950 International Congress of Mathematicians, one of the first times mathematics' highest honour recognized work built substantially around making rigorous sense of an object engineers and physicists had already been using for decades.[^schwartz1950] ## Definitions Two rigorous definitions replace the heuristic "infinite spike." As a measure, the delta assigns the value 1 to any set that contains the point 0 and the value 0 to any set that does not, so integrating a function against it in the sense of measure theory means evaluating that function at 0. As a distribution, the more common working definition in engineering, the delta is not a pointwise object at all but a linear functional acting on smooth test functions φ, defined by `delta[phi] = phi(0)`; the familiar integral notation `integral f(t)*delta(t) dt = f(0)` is a convenient shorthand for that functional's action rather than an ordinary Riemann or Lebesgue integral. The defining sifting property extends immediately to a shifted delta: `x(t) * delta(t - t0) = x(t - t0)`, so convolving any signal with a delta located at t0 does nothing but relocate the signal to t0, unchanged in every other respect.[^tib091] A worked case makes this concrete: convolving a rectangular pulse of height 4 and width 4 with a unit impulse at t = −6 simply recentres the same pulse at t = −6, `4*Pi(t/4) * delta(t + 6) = 4*Pi((t + 6)/4)`, with no change to its height or width.[^tib093] This shifting behaviour is also why the delta is the identity element for convolution: convolving any function with δ returns that function unchanged. Generalizing to n dimensions, the delta becomes a product of one-dimensional deltas along each coordinate, `delta(x,y,z) = delta(x)*delta(y)*delta(z)` in three dimensions, sifting out a function's value at a single point in space rather than on a line; a further generalization spreads the same idea over a curve or surface rather than a point, sifting out a function's values along that whole set instead of at a single one. ## Properties Several algebraic identities make the delta easy to manipulate without ever asking what it equals pointwise. It is even and symmetric, `delta(-t) = delta(t)`, and it scales as `delta(a*t) = delta(t)/abs(a)` for any nonzero a, so compressing its argument shrinks its effective width but raises its height in exact compensation, leaving the total area fixed at 1. Multiplying the delta by an ordinary continuous function collapses to a scaled delta at the same point, `f(t)*delta(t - a) = f(a)*delta(t - a)`, since the product can only be sampled where the delta itself is nonzero. Composing the delta with a differentiable function g(t) that has isolated simple zeros t_i produces a sum of deltas at those zeros, each weighted by the reciprocal of `abs(g'(t_i))`; a steeply sloped zero crossing contributes a narrower, taller term than a shallow one, because the delta's argument is changing faster near a steep zero for the same small change in t. Integrating the delta from far in the past up to a running variable t produces a function that is 0 before the delta's location and 1 after it, a single sharp step standing in for what is, in the distributional sense, the delta's own antiderivative; conversely, the delta itself is the derivative of that step, in the same distributional sense that lets a discontinuous function have a well-defined derivative at all. ## Derivatives Because the delta is defined only by its action inside an integral, its derivative is defined the same way, by integration by parts rather than by any limiting difference quotient: `integral f(t)*delta'(t) dt = -f'(0)`, and each further derivative introduces another sign flip and picks out one more order of derivative of f at the origin, `integral f(t)*delta^n(t) dt = (-1)^n * f^n(0)`. A derivative of the delta models a point dipole rather than a point source, the limit of two equal and opposite point sources brought together as their separation shrinks and their strength grows to keep the product finite. In higher dimensions the same construction applies coordinate by coordinate, so the gradient of an n-dimensional delta is a vector of one-dimensional derivative-deltas, each sifting out a different partial derivative of a test function at the origin; the construction underlies how point dipoles and point-source derivatives are represented in electromagnetism and continuum mechanics alike. ## Representations The most concrete way to represent the delta, and the one this article's primary microsim builds directly, is as a limit of a family of ordinary functions, each normalized to unit area and each narrower than the last. Three such nascent deltas are common: a Gaussian, `delta_eps(t) = (1/(eps*sqrt(2*pi))) * exp(-t^2/(2*eps^2))`; a rectangle of width ε, `delta_eps(t) = 1/eps` for `abs(t) < eps/2` and 0 otherwise; and a Lorentzian, `delta_eps(t) = (eps/pi) / (t^2 + eps^2)`. Every one of these integrates to exactly 1 for every ε greater than 0; only in the limit ε → 0, which no finite ε ever reaches, does the family's shape collapse to the idealized spike while its area stays fixed. A second representation works in frequency rather than time. The [[Fourier_transform|Fourier transform]] of the delta is the constant function 1, meaning the delta contains every frequency in equal measure, and by the same token the inverse transform of a constant is a delta: a signal with a perfectly flat spectrum is, in the time domain, an impulse. This duality is why an ideal impulse is the standard test input for measuring a system's impulse response, since it excites every frequency the system can respond to, all at once and in equal proportion. A third family of representations comes from partial differential equations whose fundamental solutions collapse to a delta as a parameter shrinks to zero, most famously the heat kernel, whose spreading Gaussian shape narrows back down to a point source as the elapsed time goes to zero. ## Dirac comb An infinite, evenly spaced train of unit deltas, `Sha(t) = sum_n delta(t - n*T)`, is called the Dirac comb, and it is the mathematical idealization of instantaneous [[Sampling_(signal_processing)|sampling]]: multiplying a continuous signal by a Dirac comb of spacing T produces a sequence of impulses, each weighted by the signal's exact value at that instant, with nothing lost from between the samples. The comb's own Fourier transform is another Dirac comb, spaced `1/T` apart in frequency rather than T apart in time, which is the reason sampling a signal in time creates repeated copies of its spectrum spaced `1/T` apart in frequency: multiplying by a comb in one domain is the same operation as convolving with a comb in the other. ## Sokhotski–Plemelj theorem A subtler identity, the Sokhotski–Plemelj theorem, connects the delta to complex analysis: as a small positive number ε shrinks to zero, `1/(x - i*eps)` approaches the principal value of `1/x` plus an imaginary term proportional to the delta, `1/(x - i*eps) -> PV(1/x) + i*pi*delta(x)`. The theorem is a standard route from a complex, analytic expression to a real distributional one, and it appears throughout scattering theory and dispersion relations, wherever a physical response function must be analytic in one half of the complex plane and its boundary value on the real axis needs to be split into a smooth part and a delta-like resonance. ## Relationship to the Kronecker delta The discrete-time counterpart of the delta is the Kronecker delta, an ordinary doubly indexed array equal to 1 when its two indices match and 0 otherwise, with no need for distribution theory at all: it is simply an array, well defined pointwise, unlike its continuous namesake. The two share the same sifting behaviour in their own settings: summing a [[Sequence|sequence]] against a Kronecker delta picks out exactly one term, the way integrating a function against the Dirac delta picks out exactly one value. The discrete version is the unit impulse against which a discrete-time [[Linear_time-invariant_system|linear time-invariant system's]] own impulse response is defined, playing the same foundational role in [[Digital_signal_processing|digital signal processing]] that the continuous delta plays in continuous time. ## Applications Three fields lean on the delta as a matter of routine. In [[Probability_theory|probability theory]], a random variable that takes one value with certainty has a [[Probability_density_function|probability density]] that is a delta at that value, which lets a discrete outcome be folded into the same density formalism built for continuous ones; a mixture of discrete and continuous outcomes is then just a sum of an ordinary density and one or more weighted deltas. In [[Quantum_mechanics|quantum mechanics]], position and momentum have continuous spectra, and their eigenstates cannot be normalized to a finite total probability the way a bound state can; the standard resolution is to normalize them instead to a delta, `<x|x'> = delta(x - x')`, meaning two position eigenstates overlap perfectly only when they coincide exactly, and not at all otherwise. The same convention extends to momentum eigenstates and underlies how a continuous basis is handled throughout the subject. In structural mechanics, a concentrated [[Force|force]] applied at one point of a beam or plate is written into the same load-distribution formalism built for a spread-out load by placing a delta at that point, `q(x) = P*delta(x - a)` for a point load of magnitude P at position a; the beam's deflection equation can then be solved once for a general distributed load, and a point load recovered simply by substituting a delta for the load density, rather than treated as a separate case requiring its own method. ## Microsims The primary microsim, *Dirac delta functions*, builds the delta the honest way, as a limit rather than a claimed object. Two sliders set the width ε of a nascent bump function and the location a where it sits; a button cycles the bump's shape through three families, Gaussian, rectangle and Lorentzian. Every family integrates to exactly 1.000 for every ε the slider allows, from a wide, gentle bump at ε = 2.0 down to an already very sharp spike at ε = 0.02, with the sketch's own defaults sitting at ε = 0.6 and the impulse placed at a = 1.5. A lower pane multiplies the current bump against a smooth test function, shades the product, and runs a fine Riemann sum on it live, so the reader can watch that running integral converge on the test function's exact value at a as ε shrinks. A three.js companion sim, built on the [[Convolution]] model, renders the same narrowing, unit-area pulse sifting a signal as it shrinks. *Try:* Narrow ε from its default of 0.6 toward 0.02 while watching the peak height climb and the shaded area hold at 1.000, then slide a across the test curve and watch the running integral track the curve's own value at every point it passes. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Dirac_delta_function) : [Wikitube](https://en.wikitube.io/wiki/Dirac_delta_function) Skeleton mirrored at revision 1358108068. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Convolution]] - [[Impulse_response]] - [[Sampling_(signal_processing)]] - [[Fourier_transform]] - [[Fourier_series]] - [[Probability_density_function]] - [[Quantum_mechanics]] - [[Linear_time-invariant_system]] - [[Digital_signal_processing]] - [[Signal_processing]] ## Notes The primary microsim's three nascent-delta families are ILLUSTRATIVE: each is an ordinary function chosen for its clean closed form, not a physical noise or signal source, and none of them ever reaches the idealized spike at any finite width the slider allows, which is the honest point the sketch is built to make. Page numbers in the references below are PDF pages of the linked open edition. ## References [^dirac1930]: Dirac, P. A. M. *The Principles of Quantum Mechanics*. Oxford: Clarendon Press, 1930 (introduces the delta function and its notation for continuous eigenstates). [^cn_history]: Citation needed: a precise account of the nineteenth-century precursors (Fourier's integral theorem, and informal impulse-like devices in the work of Cauchy, Poisson and Kirchhoff) and exactly which of them Dirac and Schwartz were aware of would need a dedicated history-of-mathematics source, not asserted here. [^schwartz1950]: Fields Medal citation, International Congress of Mathematicians, Cambridge, Massachusetts, 1950 (awarded to Laurent Schwartz for the théorie des distributions). [^tib091]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering — Theory*. 2026, p. 91 (Eq. 7.4, convolution with a shifted delta). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tib093]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering — Theory*. 2026, p. 93 (worked example, a shifted rectangular pulse). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. ## External links - Live sketch: https://editor.p5js.org/sciencenibber/full/cSVooLw3F - Editor (fork): https://editor.p5js.org/sciencenibber/sketches/cSVooLw3F <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->