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{{Short description|Generalized function whose value is zero everywhere except at zero}}
[[File:Dirac distribution PDF.svg|325px|thumb|Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.]]
[[File:Dirac function approximation.gif|right|frame|The Dirac delta as the limit as <math> a \to 0</math> (in the sense of [[Distribution (mathematics)|distribution]]s) of the sequence of zero-centered [[Normal distribution|normal distribution]]s <math>\delta_a(x) = \frac{1}{\left|a\right| \sqrt{\pi}} e^{-(x/a)^2}</math> ]]
{{Differential equations}}

In [[Mathematical analysis|mathematical analysis]], the '''Dirac delta function''' (or '''{{mvar|δ}} distribution'''), also known as the '''unit impulse''',{{sfn|atis|2013|loc=unit impulse}} is a [[Generalized function|generalized function]] on the real numbers, whose value is zero everywhere except at zero, and whose [[Integral|integral]] over the entire real line is equal to one.{{sfn|Arfken|Weber|2000|p=84}}{{sfn|Dirac|1930|loc=§22 The ''δ'' function}}{{sfn|Gelfand|Shilov|1966–1968|loc=Volume I, §1.1}}  Since there is no function having this property, to model the delta "function" rigorously involves the use of [[Limit (mathematics)|limits]] or, as is common in mathematics, measure theory and the theory of [[Distribution (mathematics)|distributions]].

The delta function was introduced by physicist [[Biography:Paul Dirac|Paul Dirac]], and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses.  It is called the delta function because it is a continuous analogue of the [[Kronecker delta]] function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was disputed until [[Biography:Laurent Schwartz|Laurent Schwartz]] developed the theory of distributions, where it is defined as a linear form acting on functions.
== Motivation and overview ==
The [[Graph of a function|graph]] of the Dirac delta is usually thought of as following the whole ''x''-axis and the positive ''y''-axis.<ref>{{Cite book|last=Zhao|first=Ji-Cheng|url={{google books |plainurl=y |id=blZYGDREpk8C|page=174}}|title=Methods for Phase Diagram Determination|date=2011-05-05|publisher=Elsevier|isbn=978-0-08-054996-5|language=en}}</ref>{{rp|174}} The Dirac delta is used to model a tall narrow spike function (an ''impulse''), and other similar [[Philosophy:Abstraction|abstraction]]s such as a point charge, point mass or [[Physics:Electron|electron]] point. For example, to calculate the [[Physics:Dynamics (mechanics)|dynamics]] of a [[Engineering:Billiard ball|billiard ball]] being struck, one can approximate the [[Physics:Force|force]] of the impact by a Dirac delta.  In doing so, one not only simplifies the equations, but one also is able to calculate the [[Physics:Motion|motion]] of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic energy transfer at subatomic levels (for instance).

To be specific, suppose that a billiard ball is at rest.  At time <math>t=0</math> it is struck by another ball, imparting it with a [[Physics:Momentum|momentum]] {{mvar|P}}, with units kg&sdot;m&sdot;s<sup>&minus;1</sup>.  The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous.  The [[Physics:Force|force]] therefore is {{math|''P'' ''δ''(''t'')}}; the units of {{math|''δ''(''t'')}} are s<sup>&minus;1</sup>.

To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval {{nowrap|<math>\Delta t = [0,T]</math>.}} That is,

<math display="block">F_{\Delta t}(t) = \begin{cases} P/\Delta t& 0<t\leq T, \\ 0 &\text{otherwise}. \end{cases}</math>

Then the momentum at any time {{mvar|t}} is found by integration:

<math display="block">p(t) = \int_0^t F_{\Delta t}(\tau)\,d\tau = \begin{cases} P & t \ge T\\ P\,t/\Delta t & 0 \le t \le T\\ 0&\text{otherwise.}\end{cases}</math>

Now, the model situation of an instantaneous transfer of momentum requires taking the limit as {{math|&Delta;''t'' &rarr; 0}}, giving a result everywhere except at {{math|0}}:

<math display="block">p(t)=\begin{cases}P & t > 0\\ 0 & t < 0.\end{cases}</math>

Here the functions <math>F_{\Delta t}</math> are thought of as useful approximations to the idea of instantaneous transfer of momentum.

The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of [[Pointwise convergence|pointwise convergence]]) <math display="inline">\lim_{\Delta t\to 0^+}F_{\Delta t}</math> is zero everywhere but a single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property

<math display="block">\int_{-\infty}^\infty F_{\Delta t}(t)\,dt = P,</math>

which holds for all {{nowrap|<math>\Delta t>0</math>,}} should continue to hold in the limit. So, in the equation {{nowrap|<math display="inline">F(t)=P\,\delta(t)=\lim_{\Delta t\to 0}F_{\Delta t}(t)</math>,}} it is understood that the limit is always taken {{em|outside the integral}}.

In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (a weak limit) of a [[Sequence|sequence]] of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with [[Variance|variance]] tending to zero.

The Dirac delta is not truly a function, at least not a usual one with domain and range in [[Real number|real number]]s. For example, the objects {{math|1=''f''(''x'') = ''δ''(''x'')}} and {{math|1=''g''(''x'') = 0}} are equal everywhere except at {{math|1=''x'' = 0}} yet have integrals that are different.  According to Lebesgue integration theory, if {{mvar|f}} and {{mvar|g}} are functions such that {{math|1=''f'' = ''g''}} [[Almost everywhere|almost everywhere]], then {{mvar|f}} is integrable [[If and only if|if and only if]] {{mvar|g}} is integrable and the integrals of {{mvar|f}} and {{mvar|g}} are identical.  A rigorous approach to regarding the Dirac delta function as a [[Philosophy:Mathematical object|mathematical object]] in its own right requires measure theory or the theory of [[Distribution (mathematics)|distribution]]s.

==History==
[[Biography:Joseph Fourier|Joseph Fourier]] presented what is now called the Fourier integral theorem in his treatise ''Théorie analytique de la chaleur'' in the form:<ref name=Fourier>{{cite book |title=The Analytical Theory of Heat |first=JB |last=Fourier |year=1822 |page=[{{google books |plainurl=y |id=-N8EAAAAYAAJ|page=408}}] |edition= English translation by Alexander Freeman, 1878 |publisher=The University Press}}, cf. {{google books |plainurl=y |id=-N8EAAAAYAAJ|page=449
}} and pp. 546–551. [{{google books |plainurl=y |id=TDQJAAAAIAAJ|page=525}} Original French text].</ref>

<math display="block">f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty\ \ d\alpha \, f(\alpha) \ \int_{-\infty}^\infty dp\ \cos  (px-p\alpha)\ , </math>

which is tantamount to the introduction of the {{mvar|δ}}-function in the form:<ref name= Kawai>{{cite book |title=Microlocal Analysis and Complex Fourier Analysis |editor1=[[Biography:Takahiro Kawai|Takahiro Kawai]] |editor2=Keiko Fujita |first=Hikosaburo |last=Komatsu |chapter=Fourier's hyperfunctions and Heaviside's pseudodifferential operators |isbn=978-981-238-161-3 |year=2002 |publisher=World Scientific |chapter-url={{google books |plainurl=y |id=8GwKzEemrIcC}}|page=[{{google books |plainurl=y |id=8GwKzEemrIcC|page=200}}]}}</ref>

<math display="block">\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty dp\ \cos  (px-p\alpha) \ . </math>

Later, Augustin Cauchy expressed the theorem using exponentials:<ref name= Myint-U>{{cite book |first1=Tyn |last1=Myint-U. |first2=Lokenath |last2=Debnath|title=Linear Partial Differential Equations for Scientists And Engineers|url={{google books |plainurl=y |id=Zbz5_UvERIIC}} |isbn=978-0-8176-4393-5 |edition=4th  |year=2007  |page=[{{google books |plainurl=y |id=Zbz5_UvERIIC|page=4}}] |publisher=Springer}}</ref><ref name=Debnath>{{cite book |title=Integral Transforms And Their Applications |first1=Lokenath |last1=Debnath |first2=Dambaru |last2=Bhatta |isbn=978-1-58488-575-7 |url={{google books |plainurl=y |id=WbZcqdvCEfwC}}|year=2007 |edition=2nd |publisher=CRC Press |page=[{{google books |plainurl=y |id=WbZcqdvCEfwC|page=2}}]}}</ref>

<math display="block">f(x)=\frac{1}{2\pi} \int_{-\infty} ^ \infty \ e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp. </math>

Cauchy pointed out that in some circumstances the ''order'' of integration is significant in this result (contrast [[Fubini's theorem]]).<ref name=Grattan-Guinness>{{cite book |title=Convolutions in French Mathematics, 1800–1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics, Volume 2 |url={{google books |plainurl=y |id=_GgioErrbW8C}}|isbn=978-3-7643-2238-0 |year=2009 |publisher=Birkhäuser |first=Ivor |last=Grattan-Guinness|page=[{{google books |plainurl=y |id=_GgioErrbW8C|page=653}} 653]}}</ref><ref name=Cauchy>

See, for example, {{Cite book|last=Cauchy|first=Augustin-Louis (1789-1857) Auteur du texte|url=https://gallica.bnf.fr/ark:/12148/bpt6k90181x|title=Oeuvres complètes d'Augustin Cauchy. Série 1, tome 1 / publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique...|date=1882–1974|language=EN|chapter-url=http://gallica.bnf.fr/ark:/12148/bpt6k90181x/f387 |chapter=Des intégrales doubles qui se présentent sous une forme indéterminèe}}</ref>

As justified using the [[Distribution (mathematics)|theory of distributions]], the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the ''δ''-function as

<math display="block">\begin{align}
f(x)&=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp \\[4pt]
&=\frac{1}{2\pi} \int_{-\infty}^\infty \left(\int_{-\infty}^\infty e^{ipx} e^{-ip\alpha } \,dp \right)f(\alpha)\,d \alpha =\int_{-\infty}^\infty \delta (x-\alpha) f(\alpha) \,d \alpha,
\end{align}</math>

where the ''δ''-function is expressed as

<math display="block">\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\,dp \ . </math>

A rigorous interpretation of the exponential form and the various limitations upon the function ''f'' necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows:<ref name="Mitrović">{{cite book |title=Fundamentals of Applied Functional Analysis: Distributions, Sobolev Spaces |first1=Dragiša |last1=Mitrović |first2=Darko |last2=Žubrinić |url={{google books |plainurl=y |id=Od5BxTEN0VsC}}|page=[{{google books |plainurl=y |id=Od5BxTEN0VsC|page=62}} 62] |isbn=978-0-582-24694-2 |year=1998 |publisher=CRC Press}}</ref>
: The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions decrease sufficiently rapidly to zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.

Further developments included generalization of the Fourier integral, "beginning with [[Biography:Michel Plancherel|Plancherel's]] pathbreaking ''L''<sup>2</sup>-theory (1910), continuing with [[Biography:Norbert Wiener|Wiener's]] and [[Biography:Salomon Bochner|Bochner's]] works (around 1930) and culminating with the amalgamation into [[Biography:Laurent Schwartz|L. Schwartz's]] theory of [[Distribution (mathematics)|distributions]] (1945)&nbsp;...",<ref name=Kracht>{{cite book |title=Topics in Mathematical Analysis: A Volume Dedicated to the Memory of A.L. Cauchy |chapter-url={{google books |plainurl=y |id=xIsPrSiDlZIC}}|first1=Manfred |last1=Kracht |first2=Erwin |last2=Kreyszig|page={{google books |plainurl=y |id=xIsPrSiDlZIC|page=553 }} 553] |isbn=978-9971-5-0666-7 |editor=Themistocles M. Rassias |year=1989 |publisher=World Scientific |chapter=On singular integral operators and generalizations}}</ref> and leading to the formal development of the Dirac delta function.

An [[Infinitesimal|infinitesimal]] formula for an infinitely tall, unit impulse delta function (infinitesimal version of [[Cauchy distribution]]) explicitly appears in an 1827 text of Augustin Louis Cauchy. {{sfn|Laugwitz|1989|p=230}} [[Biography:Siméon Denis Poisson|Siméon Denis Poisson]] considered the issue in connection with the study of wave propagation as did [[Biography:Gustav Kirchhoff|Gustav Kirchhoff]] somewhat later. Kirchhoff and [[Biography:Hermann von Helmholtz|Hermann von Helmholtz]] also introduced the unit impulse as a limit of Gaussians, which also corresponded to [[Biography:Lord Kelvin|Lord Kelvin]]'s notion of a point heat source.  At the end of the 19th century, Oliver Heaviside used formal [[Fourier series]] to manipulate the unit impulse.<ref>A more complete historical account can be found in {{harvnb|van der Pol|Bremmer|1987|loc=§V.4}}.</ref> The Dirac delta function as such was introduced by [[Biography:Paul Dirac|Paul Dirac]] in his 1927 paper ''The Physical Interpretation of the Quantum Dynamics''<ref>{{Cite journal |date=January 1927 |title=The physical interpretation of the quantum dynamics |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=113 |issue=765 |pages=621–641 |doi=10.1098/rspa.1927.0012 |bibcode=1927RSPSA.113..621D |issn=0950-1207|last1=Dirac |first1=P. A. M. |s2cid=122855515 |doi-access=free }}</ref> and used in his textbook ''[[Physics:The Principles of Quantum Mechanics|The Principles of Quantum Mechanics]]''.{{sfn|Dirac|1930|loc=§22 The ''δ'' function}}  He called it the "delta function" since he used it as a continuous analogue of the discrete [[Kronecker delta]].

==Definitions==
The Dirac delta function <math>\delta (x)</math> can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

<math display="block">\delta(x) \simeq \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}</math>

and which is also constrained to satisfy the identity{{sfn|Gelfand|Shilov|1966–1968|loc=Volume I, §1.1, p. 1}}

<math display="block">\int_{-\infty}^\infty \delta(x) \, dx = 1.</math>

This is merely a [[Heuristic|heuristic]] characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.{{sfn|Dirac|1930|p=63}}

===As a measure===
One way to rigorously capture the notion of the Dirac delta function is to define a [[Measure (mathematics)|measure]], called [[Dirac measure]], which accepts a subset {{mvar|A}} of the real line {{math|'''R'''}} as an argument, and returns {{math|1=''δ''(''A'') = 1}} if {{math|0 ∈ ''A''}}, and {{math|1=''δ''(''A'') = 0}} otherwise.<ref name="Rudin 1966 loc=§1.20">{{harvnb|Rudin|1966|loc=§1.20}}</ref>  If the delta function is conceptualized as modeling an idealized point mass at 0, then {{math|''δ''(''A'')}} represents the mass contained in the set {{mvar|A}}.  One may then define the integral against {{mvar|δ}} as the integral of a function against this mass distribution.  Formally, the Lebesgue integral provides the necessary analytic device. The Lebesgue integral with respect to the measure {{mvar|δ}} satisfies

<math display="block">\int_{-\infty}^\infty f(x) \, \delta(dx) =  f(0)</math>

for all continuous compactly supported functions {{mvar|f}}. The measure {{mvar|δ}} is not absolutely continuous with respect to the [[Lebesgue measure]]—in fact, it is a [[Singular measure|singular measure]]. Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which the property

<math display="block">\int_{-\infty}^\infty f(x)\, \delta(x)\, dx = f(0)</math>

holds.{{sfn|Hewitt|Stromberg|1963|loc=§19.61}} As a result, the latter notation is a convenient [[Abuse of notation|abuse of notation]], and not a standard ([[Riemann integral|Riemann]] or Lebesgue) integral.

As a [[Probability measure|probability measure]] on {{math|'''R'''}}, the delta measure is characterized by its [[Cumulative distribution function|cumulative distribution function]], which is the unit step function.<ref>{{harvnb|Driggers|2003|p=2321}}  See also {{harvnb|Bracewell|1986|loc=Chapter 5}} for a different interpretation.  Other conventions for the assigning the value of the Heaviside function at zero exist, and some of these are not consistent with what follows.</ref>

<math display="block">H(x) = 
\begin{cases}
1 & \text{if } x\ge 0\\
0 & \text{if } x < 0.
\end{cases}</math>

This means that {{math|''H''(''x'')}} is the integral of the cumulative [[Indicator function|indicator function]] {{math|'''1'''<sub>(−∞, ''x'']</sub>}} with respect to the measure {{mvar|δ}}; to wit,

<math display="block">H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta(dt) = \delta(-\infty,x],</math>

the latter being the measure of this interval; more formally, {{math|''δ''((−∞, ''x''])}}. Thus in particular the integration of the delta function against a continuous function can be properly understood as a [[Riemann–Stieltjes integral]]:{{sfn|Hewitt|Stromberg|1963|loc=§9.19}}

<math display="block">\int_{-\infty}^\infty f(x)\,\delta(dx) = \int_{-\infty}^\infty f(x) \,dH(x).</math>

All higher [[Moment (mathematics)|moments]] of {{mvar|δ}} are zero.  In particular, [[Characteristic function (probability theory)|characteristic function]] and moment generating function are both equal to one.

===As a distribution===
In the theory of [[Distribution (mathematics)|distributions]], a generalized function is considered not a function in itself but only through how it affects other functions when "integrated" against them.{{sfn|Hazewinkel|2011|p=[{{google books |plainurl=y |id=_YPtCAAAQBAJ|page=41}} 41]}} In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function is against a sufficiently "good" '''test function''' {{mvar|φ}}.  Test functions are also known as [[Bump function|bump function]]s.  If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral.

A typical space of test functions consists of all smooth functions on {{math|'''R'''}} with compact support that have as many derivatives as required.  As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by{{sfn|Strichartz|1994|loc=§2.2}}

{{NumBlk|:| <math>\delta[\varphi] = \varphi(0)</math>|{{EquationRef|1}}}}

for every test function {{mvar|φ}}.

For {{mvar|δ}} to be properly a distribution, it must be continuous in a suitable topology on the space of test functions.  In general, for a linear functional {{mvar|S}} on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer {{mvar|N}} there is an integer {{math|''M''<sub>''N''</sub>}} and a constant {{mvar|''C''<sub>''N''</sub>}} such that for every test function {{mvar|φ}}, one has the inequality{{sfn|Hörmander|1983|loc=Theorem 2.1.5}}

<math display="block">\left|S[\varphi]\right| \le C_N \sum_{k=0}^{M_N}\sup_{x\in [-N,N]} \left|\varphi^{(k)}(x)\right|</math>

where {{math|sup}} represents the [[Infimum and supremum|supremum]]. With the {{mvar|δ}} distribution, one has such an inequality (with {{math|1=''C''<sub>''N''</sub> = 1)}} with {{math|1=''M''<sub>''N''</sub> = 0}} for all {{mvar|N}}.  Thus {{mvar|δ}} is a distribution of order zero. It is, furthermore, a distribution with compact support (the [[Support (mathematics)|support]] being {{math|{{brace|0}}}}).

The delta distribution can also be defined in several equivalent ways.  For instance, it is the distributional derivative of the [[Heaviside step function]]. This means that for every test function {{mvar|φ}}, one has

<math display="block">\delta[\varphi] = -\int_{-\infty}^\infty \varphi'(x)\,H(x)\,dx.</math>

Intuitively, if [[Integration by parts|integration by parts]] were permitted, then the latter integral should simplify to

<math display="block">\int_{-\infty}^\infty \varphi(x)\,H'(x)\,dx = \int_{-\infty}^\infty \varphi(x)\,\delta(x)\,dx,</math>

and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have

<math display="block">-\int_{-\infty}^\infty \varphi'(x)\,H(x)\, dx = \int_{-\infty}^\infty \varphi(x)\,dH(x).</math>

In the context of measure theory, the Dirac measure gives rise to distribution by integration.  Conversely, equation ({{EquationNote|1}}) defines a [[Daniell integral]] on the space of all compactly supported continuous functions {{mvar|φ}} which, by the [[Riesz–Markov–Kakutani representation theorem|Riesz representation theorem]], can be represented as the Lebesgue integral of {{mvar|φ}} with respect to some [[Radon measure]].

Generally, when the term ''Dirac delta function'' is used, it is in the sense of distributions rather than measures, the [[Dirac measure]] being among several terms for the corresponding notion in measure theory.  Some sources may also use the term ''Dirac delta distribution''.

===Generalizations===
The delta function can be defined in {{mvar|n}}-dimensional [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} as the measure such that

<math display="block">\int_{\mathbf{R}^n} f(\mathbf{x})\,\delta(d\mathbf{x}) = f(\mathbf{0})</math>

for every compactly supported continuous function {{mvar|f}}.  As a measure, the {{mvar|n}}-dimensional delta function is the [[Product measure|product measure]] of the 1-dimensional delta functions in each variable separately. Thus, formally, with {{math|1='''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>)}}, one has{{sfn|Bracewell|1986|loc=Chapter 5}}

{{NumBlk|:|<math>\delta(\mathbf{x}) = \delta(x_1)\,\delta(x_2)\cdots\delta(x_n).</math>|{{EquationRef|2}}}}

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.{{sfn|Hörmander|1983|loc=§3.1}}  However, despite widespread use in engineering contexts, ({{EquationNote|2}}) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.{{sfn|Strichartz|1994|loc=§2.3}}{{sfn|Hörmander|1983|loc=§8.2}}

The notion of a '''[[Dirac measure]]''' makes sense on any set.{{sfn|Rudin |1966 |loc=§1.20}} Thus if {{mvar|X}} is a set, {{math|''x''<sub>0</sub> ∈ ''X''}} is a marked point, and {{math|Σ}} is any sigma algebra of subsets of {{mvar|X}}, then the measure defined on sets {{math|''A'' ∈ Σ}} by

<math display="block">\delta_{x_0}(A)=\begin{cases}
1 &\text{if }x_0\in A\\
0 &\text{if }x_0\notin A
\end{cases}</math>

is the delta measure or unit mass concentrated at {{math|''x''<sub>0</sub>}}.

Another common generalization of the delta function is to a [[Differentiable manifold|differentiable manifold]] where most of its properties as a distribution can also be exploited because of the differentiable structure.  The delta function on a manifold {{mvar|M}} centered at the point {{math|''x''<sub>0</sub> ∈ ''M''}} is defined as the following distribution:

{{NumBlk|:|<math>\delta_{x_0}[\varphi] = \varphi(x_0)</math>|{{EquationRef|3}}}}

for all compactly supported smooth real-valued functions {{mvar|φ}} on {{mvar|M}}.{{sfn|Dieudonné|1972|loc=§17.3.3}} A common special case of this construction is a case in which {{mvar|M}} is an [[Open set|open set]] in the Euclidean space {{math|'''R'''<sup>''n''</sup>}}.

On a locally compact Hausdorff space {{mvar|X}}, the Dirac delta measure concentrated at a point {{mvar|x}} is the [[Radon measure]] associated with the Daniell integral ({{EquationNote|3}}) on compactly supported continuous functions {{mvar|φ}}.<ref>{{Cite book|last1=Krantz|first1=Steven G.|url={{google books |plainurl=y |id=X_BKmVphIcsC&q }}|title=Geometric Integration Theory|last2=Parks|first2=Harold R.|date=2008-12-15|publisher=Springer Science & Business Media|isbn=978-0-8176-4679-0|language=en}}</ref> At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping <math>x_0\mapsto \delta_{x_0}</math> is a continuous embedding of {{mvar|X}} into the space of finite Radon measures on {{mvar|X}}, equipped with its [[Vague topology|vague topology]]. Moreover, the [[Convex hull|convex hull]] of the image of {{mvar|X}} under this embedding is [[Dense set|dense]] in the space of probability measures on {{mvar|X}}.{{sfn|Federer|1969|loc=§2.5.19}}

==Properties==

===Scaling and symmetry===
The delta function satisfies the following scaling property for a non-zero scalar {{mvar|α}}:{{sfn|Strichartz|1994|loc=Problem 2.6.2}}

<math display="block">\int_{-\infty}^\infty \delta(\alpha x)\,dx
=\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|}
=\frac{1}{|\alpha|}</math>

and so
{{NumBlk|:|<math>\delta(\alpha x) = \frac{\delta(x)}{|\alpha|}.</math>|{{EquationRef|4}}}}

Scaling property proof:
<math display="block">\int\limits_{-\infty}^{\infty} dx\ g(x) \delta (ax) = \frac{1}{a}\int\limits_{-\infty}^{\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x')
</math>
where a change of variable {{math|1=''x&prime;'' = ''ax''}} is used. If {{mvar|a}} is negative, i.e., {{math|1=''a'' = &minus;{{!}}''a''{{!}}}}, then
<math display="block">\int\limits_{-\infty}^{\infty} dx\ g(x) \delta (ax) = \frac{1}{-\left \vert a \right \vert}\int\limits_{\infty}^{-\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x') = \frac{1}{\left \vert a \right \vert}\int\limits_{-\infty}^{\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x') = \frac{1}{\left \vert a \right \vert}g(0).
</math>
Thus, {{nowrap|<math>\delta (ax) = \frac{1}{\left \vert a \right \vert} \delta(x)</math>.}}

In particular, the delta function is an even distribution (symmetry), in the sense that

<math display="block">\delta(-x) = \delta(x)</math>

which is [[Homogeneous function|homogeneous]] of degree {{math|&minus;1}}.

===Algebraic properties===
The [[Distribution (mathematics)|distributional product]] of {{mvar|δ}} with {{mvar|x}} is equal to zero:

<math display="block">x\,\delta(x) = 0.</math>

More generally, <math>(x-a)^n\delta(x-a) =0</math> for all positive integers <math>n</math>.

Conversely, if {{math|1=''xf''(''x'') = ''xg''(''x'')}}, where {{mvar|f}} and {{mvar|g}} are distributions, then

<math display="block">f(x) = g(x) +c \delta(x)</math>

for some constant {{mvar|c}}.{{sfn|Vladimirov|1971|loc=Chapter 2, Example 3(d)}}

===Translation===
The integral of the time-delayed Dirac delta is<ref>{{Cite book|last1=Rottwitt|first1=Karsten|url={{google books |plainurl=y |id=G1jSBQAAQBAJ}}|title=Nonlinear Optics: Principles and Applications|last2=Tidemand-Lichtenberg|first2=Peter| date=2014-12-11| publisher=CRC Press|isbn=978-1-4665-6583-8|language=en|page=[{{google books |plainurl=y |id=G1jSBQAAQBAJ|page=276}}] 276}}</ref>

<math display="block">\int_{-\infty}^\infty f(t) \,\delta(t-T)\,dt = f(T).</math>

This is sometimes referred to as the ''sifting property''<ref>{{MathWorld|urlname=SiftingProperty|title=Sifting Property}}</ref> or the ''sampling property''.<ref>{{Cite book|last=Karris|first=Steven T.|url={{google books |plainurl=y |id=f0RdM1zv_dkC}}| title=Signals and Systems with MATLAB Applications|date=2003|publisher=Orchard Publications|isbn=978-0-9709511-6-8|language=en| page=[{{google books |plainurl=y |id=f0RdM1zv_dkC&pg=SA1-PA15 }} 15]}}</ref> The delta function is said to "sift out" the value of ''f(t)'' at ''t'' = ''T''.<ref>{{Cite book|last=Roden|first=Martin S.|url={{google books |plainurl=y |id=YEKeBQAAQBAJ}}|title=Introduction to Communication Theory|date=2014-05-17|publisher=Elsevier|isbn=978-1-4831-4556-3|language=en|page=[{{google books |plainurl=y |id=YEKeBQAAQBAJ|page=40}}]}}</ref>

It follows that the effect of [[Convolution|convolving]] a function {{math|''f''(''t'')}} with the time-delayed Dirac delta <math> \delta_T(t) = \delta(t-T)</math> is to time-delay {{math|''f''(''t'')}} by the same amount:

<math display="block">\begin{align}
(f * \delta_T)(t) \ &\stackrel{\mathrm{def}}{=}\  \int_{-\infty}^\infty f(\tau)\, \delta(t-T-\tau) \, d\tau \\
&= \int_{-\infty}^\infty f(\tau)  \,\delta(\tau-(t-T)) \,d\tau \qquad \text{since}~ \delta(-x) = \delta(x) ~~ \text{by (4)}\\
&= f(t-T).
\end{align}</math>

The sifting property holds under the precise condition that {{mvar|f}} be a [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]] (see the discussion of the Fourier transform below). As a special case, for instance, we have the identity (understood in the distribution sense)

<math display="block">\int_{-\infty}^\infty \delta (\xi-x) \delta(x-\eta) \,dx = \delta(\eta-\xi).</math>

===Composition with a function===
More generally, the delta distribution may be [[Distribution (mathematics)#Composition with a smooth function|composed]] with a smooth function {{math|''g''(''x'')}} in such a way that the familiar change of variables formula holds, that

<math display="block">\int_{\R} \delta\bigl(g(x)\bigr) f\bigl(g(x)\bigr) \left|g'(x)\right| dx = \int_{g(\R)} \delta(u)\,f(u)\,du</math>

provided that {{mvar|g}} is a continuously differentiable function with {{math|''g&prime;''}} nowhere zero.{{sfn|Gelfand|Shilov|1966–1968|loc=Vol. 1, §II.2.5}} That is, there is a unique way to assign meaning to the distribution <math>\delta\circ g</math> so that this identity holds for all compactly supported test functions {{mvar|f}}.  Therefore, the domain must be broken up to exclude the {{math|1=''g&prime;'' = 0}} point.  This distribution satisfies {{math|1=''δ''(''g''(''x'')) = 0}} if {{mvar|g}} is nowhere zero, and otherwise if {{mvar|g}} has a real [[Root of a function|root]] at {{math|''x''<sub>0</sub>}}, then

<math display="block">\delta(g(x)) = \frac{\delta(x-x_0)}{|g'(x_0)|}.</math>

It is natural therefore to {{em|define}} the composition {{math|''δ''(''g''(''x''))}} for continuously differentiable functions {{mvar|g}} by

<math display="block">\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}</math>

where the sum extends over all roots (i.e., all the different ones) of {{mvar|''g''(''x'')}}, which are assumed to be simple.  Thus, for example

<math display="block">\delta\left(x^2-\alpha^2\right) = \frac{1}{2|\alpha|} \Big[\delta\left(x+\alpha\right)+\delta\left(x-\alpha\right)\Big].</math>

In the integral form, the generalized scaling property may be written as

<math display="block"> \int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}. </math>

===Indefinite integral===
For a constant <math>a \isin \mathbb{R}</math> and a "well-behaved" arbitrary real-valued function {{math|''y''(''x'')}},
<math display="block">\displaystyle{\int}y(x)\delta(x-a)dx = y(a)H(x-a) + c,</math>
where {{math|''H''(''x'')}} is the [[Heaviside step function]] and {{math|''c''}} is an integration constant.

===Properties in ''n'' dimensions===
The delta distribution in an {{mvar|n}}-dimensional space satisfies the following scaling property instead,
<math display="block">\delta(\alpha\mathbf{x}) = |\alpha|^{-n}\delta(\mathbf{x}) ~,</math>
so that {{mvar|δ}} is a [[Homogeneous function|homogeneous]] distribution of degree {{math|&minus;''n''}}.

Under any [[Reflection (mathematics)|reflection]] or [[Rotation (mathematics)|rotation]] {{mvar|ρ}}, the delta function is invariant,
<math display="block">\delta(\rho \mathbf{x}) = \delta(\mathbf{x})~.</math>

As in the one-variable case, it is possible to define the composition of {{mvar|δ}} with a bi-Lipschitz function<ref>Further refinement is possible, namely to [[Submersion (mathematics)|submersions]], although these require a more involved change of variables formula.</ref> {{math|''g'': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} uniquely so that the identity
<math display="block">\int_{\R^n} \delta(g(\mathbf{x}))\, f(g(\mathbf{x}))\left|\det g'(\mathbf{x})\right| d\mathbf{x} = \int_{g(\R^n)} \delta(\mathbf{u}) f(\mathbf{u})\,d\mathbf{u}</math>
for all compactly supported functions {{mvar|f}}.

Using the [[Coarea formula|coarea formula]] from [[Geometric measure theory|geometric measure theory]], one can also define the composition of the delta function with a [[Submersion (mathematics)|submersion]] from one Euclidean space to another one of different dimension; the result is a type of [[Current (mathematics)|current]]. In the special case of a continuously differentiable function {{math|''g'' : '''R'''<sup>''n''</sup> → '''R'''}} such that the [[Gradient|gradient]] of {{mvar|g}} is nowhere zero, the following identity holds{{sfn|Hörmander|1983|loc=§6.1}}
<math display="block">\int_{\R^n} f(\mathbf{x}) \, \delta(g(\mathbf{x})) \,d\mathbf{x} = \int_{g^{-1}(0)}\frac{f(\mathbf{x})}{|\mathbf{\nabla}g|}\,d\sigma(\mathbf{x}) </math>
where the integral on the right is over {{math|''g''<sup>−1</sup>(0)}}, the {{math|(''n'' − 1)}}-dimensional surface defined by {{math|1=''g''('''x''') = 0}} with respect to the [[Minkowski content]] measure.  This is known as a ''simple layer'' integral.

More generally, if {{mvar|S}} is a smooth hypersurface of {{math|'''R'''<sup>''n''</sup>}}, then we can associate to {{mvar|S}} the distribution that integrates any compactly supported smooth function {{mvar|g}} over {{mvar|S}}:
<math display="block">\delta_S[g] = \int_S g(\mathbf{s})\,d\sigma(\mathbf{s})</math>

where {{mvar|σ}} is the hypersurface measure associated to {{mvar|S}}.  This generalization is associated with the [[Potential theory|potential theory]] of simple layer potentials on {{mvar|S}}.  If {{mvar|D}} is a [[Domain (mathematical analysis)|domain]] in {{math|'''R'''<sup>''n''</sup>}} with smooth boundary {{mvar|S}}, then {{math|''δ''<sub>''S''</sub>}} is equal to the normal derivative of the [[Indicator function|indicator function]] of {{mvar|D}} in the distribution sense,

<math display="block">-\int_{\R^n}g(\mathbf{x})\,\frac{\partial 1_D(\mathbf{x})}{\partial n}\,d\mathbf{x}=\int_S\,g(\mathbf{s})\, d\sigma(\mathbf{s}),</math>

where {{mvar|n}} is the outward normal.{{sfn|Lange|2012|loc=pp.29–30}}{{sfn|Gelfand|Shilov|1966–1968|p=212}} For a proof, see e.g. the article on the [[Laplacian of the indicator#Dirac surface delta function|surface delta function]].

In three dimensions, the delta function is represented in spherical coordinates by:

<math display="block">\delta(\mathbf{r}-\mathbf{r}_0) = 
\begin{cases} 
    \displaystyle\frac{1}{r^2\sin\theta}\delta(r-r_0) \delta(\theta-\theta_0)\delta(\phi-\phi_0)& x_0,y_0,z_0 \ne 0 \\
    \displaystyle\frac{1}{2\pi r^2\sin\theta}\delta(r-r_0) \delta(\theta-\theta_0)& x_0=y_0=0,\ z_0 \ne 0 \\ 
    \displaystyle\frac{1}{4\pi r^2}\delta(r-r_0) & x_0=y_0=z_0 = 0  
\end{cases}</math>

==Fourier transform==
The delta function is a [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]], and therefore it has a well-defined [[Fourier transform]].  Formally, one finds<ref>The numerical factors depend on the [[Fourier transform#Other conventions|conventions]] for the Fourier transform.</ref>

<math display="block">\widehat{\delta}(\xi)=\int_{-\infty}^\infty e^{-2\pi i x \xi} \,\delta(x)dx = 1.</math>

Properly speaking, the Fourier transform of a distribution is defined by imposing [[Self-adjoint|self-adjoint]]ness of the Fourier transform under the duality pairing <math>\langle\cdot,\cdot\rangle</math> of tempered distributions with Schwartz functions.  Thus <math>\widehat{\delta}</math> is defined as the unique tempered distribution satisfying

<math display="block">\langle\widehat{\delta},\varphi\rangle = \langle\delta,\widehat{\varphi}\rangle</math>

for all Schwartz functions {{mvar|φ}}. And indeed it follows from this that <math>\widehat{\delta}=1.</math>

As a result of this identity, the [[Convolution|convolution]] of the delta function with any other tempered distribution {{mvar|S}} is simply {{mvar|S}}:

<math display="block">S*\delta = S.</math>

That is to say that {{mvar|δ}} is an [[Identity element|identity element]] for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an [[Associative algebra|associative algebra]] with identity the delta function.  This property is fundamental in [[Signal processing|signal processing]], as convolution with a tempered distribution is a [[Linear time-invariant system|linear time-invariant system]], and applying the linear time-invariant system measures its [[Impulse response|impulse response]].  The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for {{mvar|δ}}, and once it is known, it characterizes the system completely. See {{section link | LTI system theory |Impulse response and convolution}}.

The inverse Fourier transform of the tempered distribution {{math|1=''f''(''ξ'') = 1}} is the delta function. Formally, this is expressed as
<math display="block">\int_{-\infty}^\infty 1 \cdot e^{2\pi i x\xi}\,d\xi = \delta(x)</math>
and more rigorously, it follows since
<math display="block">\langle 1, \widehat{f}\rangle = f(0) = \langle\delta,f\rangle</math>
for all Schwartz functions {{mvar|''f''}}.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on {{math|'''R'''}}.  Formally, one has
<math display="block">\int_{-\infty}^\infty e^{i 2\pi \xi_1 t} \left[e^{i 2\pi \xi_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (\xi_2 - \xi_1) t} \,dt = \delta(\xi_2 - \xi_1).</math>

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution
<math display="block">f(t) = e^{i2\pi\xi_1 t}</math>
is
<math display="block">\widehat{f}(\xi_2) = \delta(\xi_1-\xi_2)</math>
which again follows by imposing self-adjointness of the Fourier transform.

By [[Analytic continuation|analytic continuation]] of the Fourier transform, the [[Laplace transform]] of the delta function is found to be{{sfn|Bracewell|1986}}
<math display="block"> \int_{0}^{\infty}\delta(t-a)\,e^{-st} \, dt=e^{-sa}.</math>

==Derivatives==
The derivative of the Dirac delta distribution, denoted {{math|''δ&prime;''}} and also called the ''Dirac delta prime'' or ''Dirac delta derivative'' as described in [[Laplacian of the indicator]], is defined on compactly supported smooth test functions {{mvar|φ}} by{{sfn|Gelfand|Shilov|1966–1968|p=26}}
<math display="block">\delta'[\varphi] = -\delta[\varphi']=-\varphi'(0).</math>

The first equality here is a kind of [[Integration by parts|integration by parts]], for if {{mvar|δ}} were a true function then
<math display="block">\int_{-\infty}^\infty \delta'(x)\varphi(x)\,dx
= \delta(x)\varphi(x)|_{-\infty}^{\infty} -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx
= -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx
= -\varphi'(0).</math>

By [[Mathematical induction|mathematical induction]], the {{mvar|k}}-th derivative of {{mvar|δ}} is defined similarly as the distribution given on test functions by

<math display="block">\delta^{(k)}[\varphi] = (-1)^k \varphi^{(k)}(0).</math>

In particular, {{mvar|δ}} is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:{{sfn|Gelfand|Shilov|1966–1968|loc=§2.1}}
<math display="block">\delta'(x) = \lim_{h\to 0} \frac{\delta(x+h)-\delta(x)}{h}.</math>

More properly, one has
<math display="block">\delta' = \lim_{h\to 0} \frac{1}{h}(\tau_h\delta - \delta)</math>
where {{mvar|τ<sub>h</sub>}} is the translation operator, defined on functions by {{math|1=''τ<sub>h</sub>φ''(''x'') = ''φ''(''x'' + ''h'')}}, and on a distribution {{mvar|S}} by
<math display="block">(\tau_h S)[\varphi] = S[\tau_{-h}\varphi].</math>

In the theory of [[Physics:Electromagnetism|electromagnetism]], the first derivative of the delta function represents a point magnetic [[Physics:Dipole|dipole]] situated at the origin.  Accordingly, it is referred to as a dipole or the [[Unit doublet|doublet function]].<ref>{{MathWorld|title=Doublet Function|urlname=DoubletFunction}}</ref>

The derivative of the delta function satisfies a number of basic properties, including:{{sfn|Bracewell|2000|p=86}}
<math display="block">
\begin{align}
 \delta'(-x) &= -\delta'(x) \\
 x\delta'(x) &= -\delta(x)
\end{align}
</math>
which can be shown by applying a test function and integrating by parts.

The latter of these properties can also be demonstrated by applying distributional derivative definition, Liebniz&nbsp;'s theorem and linearity of inner product:<ref>{{Cite web|url=https://www.matematicamente.it/forum/viewtopic.php?f=36&t=62388&start=10#wrap|title=Gugo82's comment on the distributional derivative of Dirac's delta|date=12 September 2010|website=matematicamente.it}}</ref>

<math display="block">
\begin{align}
\langle x\delta', \varphi \rangle \, &= \, \langle \delta', x\varphi \rangle \, = \, -\langle\delta,(x\varphi)'\rangle \, = \, - \langle \delta, x'\varphi + x\varphi'\rangle  \, = \, - \langle \delta, x'\varphi\rangle  - \langle\delta, x\varphi'\rangle  \, =  \,
- x'(0)\varphi(0) - x(0)\varphi'(0) \\
&= \, -x'(0) \langle \delta , \varphi \rangle - x(0) \langle \delta, \varphi' \rangle \, = \, -x'(0) \langle \delta,\varphi\rangle + x(0) \langle \delta', \varphi \rangle \, = \,
\langle x(0)\delta' - x'(0)\delta, \varphi \rangle \\
 \Longrightarrow x(t)\delta'(t) &= x(0)\delta'(t) - x'(0)\delta(t) = -x'(0)\delta(t) = -\delta(t)
\end{align}
</math>

Furthermore, the convolution of {{mvar|δ&prime;}} with a compactly-supported, smooth function {{mvar|f}} is

<math display="block">\delta'*f = \delta*f' = f',</math>

which follows from the properties of the distributional derivative of a convolution.

===Higher dimensions===
More generally, on an [[Open set|open set]] {{mvar|U}} in the {{mvar|n}}-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math>, the Dirac delta distribution centered at a point {{math|''a'' ∈ ''U''}} is defined by{{sfn|Hörmander|1983|p=56}}
<math display="block">\delta_a[\varphi]=\varphi(a)</math>
for all <math>\varphi \in C_c^\infty(U)</math>, the space of all smooth functions with compact support on {{mvar|U}}.  If <math>\alpha = (\alpha_1, \ldots, \alpha_n)</math> is any multi-index with <math> |\alpha|=\alpha_1+\cdots+\alpha_n</math> and <math>\partial^\alpha</math> denotes the associated mixed [[Partial derivative|partial derivative]] operator, then the {{mvar|α}}-th derivative {{mvar|∂<sup>α</sup>δ<sub>a</sub>}} of {{mvar|δ<sub>a</sub>}} is given by{{sfn|Hörmander|1983|p=56}}

<math display="block">\left\langle \partial^\alpha \delta_{a}, \, \varphi \right\rangle = (-1)^{| \alpha |} \left\langle \delta_{a}, \partial^{\alpha} \varphi \right\rangle = (-1)^{| \alpha |} \partial^\alpha \varphi (x) \Big|_{x = a} \quad \text{ for all } \varphi \in C_c^\infty(U).</math>

That is, the {{mvar|α}}-th derivative of {{mvar|δ<sub>a</sub>}} is the distribution whose value on any test function {{mvar|φ}} is the {{mvar|α}}-th derivative of {{mvar|φ}} at {{mvar|a}} (with the appropriate positive or negative sign).

The first partial derivatives of the delta function are thought of as [[Physics:Double layer potential|double layers]] along the coordinate planes.  More generally, the normal derivative of a simple layer supported on a surface is a double layer supported on that surface and represents a laminar magnetic monopole.  Higher derivatives of the delta function are known in physics as multipoles.

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support.  If {{mvar|S}} is any distribution on {{mvar|U}} supported on the set {{math|{{brace|''a''}}}} consisting of a single point, then there is an integer {{mvar|m}} and coefficients {{mvar|c<sub>α</sub>}} such that{{sfn|Hörmander|1983|p=56}}{{sfn|Rudin|1991|loc=Theorem 6.25}}
<math display="block">S = \sum_{|\alpha|\le m} c_\alpha \partial^\alpha\delta_a.</math>

==Representations of the delta function==
The delta function can be viewed as the limit of a sequence of functions

<math display="block">\delta (x) = \lim_{\varepsilon\to 0^+} \eta_\varepsilon(x), </math>

where {{math|''η<sub>ε</sub>''(''x'')}} is sometimes called a '''nascent delta function'''{{anchor|nascent delta function}}. This limit is meant in a weak sense: either that

{{NumBlk|:|<math> \lim_{\varepsilon\to 0^+} \int_{-\infty}^\infty \eta_\varepsilon(x)f(x) \, dx = f(0) </math>|{{EquationRef|5}}}}

for all [[Continuous function|continuous]] functions {{mvar|f}} having compact support, or that this limit holds for all smooth functions {{mvar|f}} with compact support.  The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the [[Vague topology|vague topology]] of measures, and the latter is convergence in the sense of [[Distribution (mathematics)|distributions]].

===Approximations to the identity===
Typically a nascent delta function {{mvar|η<sub>ε</sub>}} can be constructed in the following manner.  Let {{mvar|η}} be an absolutely integrable function on {{math|'''R'''}} of total integral {{math|1}}, and define
<math display="block">\eta_\varepsilon(x) = \varepsilon^{-1} \eta \left (\frac{x}{\varepsilon} \right). </math>

In {{mvar|n}} dimensions, one uses instead the scaling
<math display="block">\eta_\varepsilon(x) = \varepsilon^{-n} \eta \left (\frac{x}{\varepsilon} \right). </math>

Then a simple change of variables shows that {{mvar|η<sub>ε</sub>}} also has integral {{math|1}}.  One may show that ({{EquationNote|5}}) holds for all continuous compactly supported functions {{mvar|f}},{{sfn|Stein|Weiss|1971|loc=Theorem 1.18}} and so {{mvar|η<sub>ε</sub>}} converges weakly to {{mvar|δ}} in the sense of measures.

The {{mvar|η<sub>ε</sub>}} constructed in this way are known as an '''approximation to the identity'''.{{sfn|Rudin|1991|loc=§II.6.31}} This terminology is because the space {{math|''L''<sup>1</sup>('''R''')}} of absolutely integrable functions is closed under the operation of [[Convolution|convolution]] of functions: {{math|''f'' ∗ ''g'' ∈ ''L''<sup>1</sup>('''R''')}} whenever {{mvar|f}} and {{mvar|g}} are in {{math|''L''<sup>1</sup>('''R''')}}.  However, there is no identity in {{math|''L''<sup>1</sup>('''R''')}} for the convolution product: no element {{mvar|h}} such that {{math|1=''f'' ∗ ''h'' = ''f''}} for all {{mvar|f}}.  Nevertheless, the sequence {{mvar|η<sub>ε</sub>}} does approximate such an identity in the sense that

<math display="block">f*\eta_\varepsilon \to f \quad \text{as }\varepsilon\to 0.</math>

This limit holds in the sense of mean convergence (convergence in {{math|''L''<sup>1</sup>}}).  Further conditions on the {{mvar|η<sub>ε</sub>}}, for instance that it be a mollifier associated to a compactly supported function,<ref>More generally, one only needs {{math|1=''η'' = ''η''<sub>1</sub>}} to have an integrable radially symmetric decreasing rearrangement.</ref> are needed to ensure pointwise convergence [[Almost everywhere|almost everywhere]].

If the initial {{math|1=''η'' = ''η''<sub>1</sub>}} is itself smooth and compactly supported then the sequence is called a [[Mollifier|mollifier]].  The standard mollifier is obtained by choosing {{mvar|η}} to be a suitably normalized [[Bump function|bump function]], for instance

<math display="block">\eta(x) = \begin{cases}
e^{-\frac{1}{1-|x|^2}}& \text{if } |x| < 1\\
0                     & \text{if } |x|\geq 1.
\end{cases}</math>

In some situations such as [[Numerical analysis|numerical analysis]], a [[Piecewise linear function|piecewise linear]] approximation to the identity is desirable.  This can be obtained by taking {{math|''η''<sub>1</sub>}} to be a hat function.  With this choice of {{math|''η''<sub>1</sub>}}, one has

<math display="block"> \eta_\varepsilon(x) = \varepsilon^{-1}\max \left (1-\left|\frac{x}{\varepsilon}\right|,0 \right) </math>

which are all continuous and compactly supported, although not smooth and so not a mollifier.

===Probabilistic considerations===
In the context of [[Probability theory|probability theory]], it is natural to impose the additional condition that the initial {{math|''η''<sub>1</sub>}} in an approximation to the identity should be positive, as such a function then represents a [[Probability distribution|probability distribution]].  Convolution with a probability distribution is sometimes favorable because it does not result in [[Overshoot (signal)|overshoot]] or undershoot, as the output is a [[Convex combination|convex combination]] of the input values, and thus falls between the maximum and minimum of the input function.  Taking {{math|''η''<sub>1</sub>}} to be any probability distribution at all, and letting {{math|1=''η<sub>ε</sub>''(''x'') = ''η''<sub>1</sub>(''x''/''ε'')/''ε''}} as above will give rise to an approximation to the identity.  In general this converges more rapidly to a delta function if, in addition, {{mvar|η}} has mean {{math|0}} and has small higher moments. For instance, if {{math|''η''<sub>1</sub>}} is the [[Uniform distribution (continuous)|uniform distribution]] on {{nowrap|1=<math display="inline">\left[-\frac{1}{2},\frac{1}{2}\right]</math>,}} also known as the [[Rectangular function|rectangular function]], then:{{sfn|Saichev|Woyczyński|1997|loc=§1.1 The "delta function" as viewed by a physicist and an engineer, p. 3}}
<math display="block">
\eta_\varepsilon(x) = \frac{1}{\varepsilon}\operatorname{rect}\left(\frac{x}{\varepsilon}\right)=
\begin{cases}
\frac{1}{\varepsilon},&-\frac{\varepsilon}{2}<x<\frac{\varepsilon}{2}, \\
0,                    &\text{otherwise}.
\end{cases}</math>

Another example is with the [[Wigner semicircle distribution]]
<math display="block">\eta_\varepsilon(x)= \begin{cases}
\frac{2}{\pi \varepsilon^2}\sqrt{\varepsilon^2 - x^2}, & -\varepsilon < x < \varepsilon, \\
0, & \text{otherwise}.
\end{cases}</math>

This is continuous and compactly supported, but not a mollifier because it is not smooth.

===Semigroups===
Nascent delta functions often arise as convolution [[Semigroup|semigroup]]s.<ref>{{Cite book|last1=Milovanović|first1=Gradimir V.|url={{google books |plainurl=y |id=4U-5BQAAQBAJ}}|title=Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava|last2=Rassias|first2=Michael Th|date=2014-07-08|publisher=Springer|isbn=978-1-4939-0258-3|language=en|page=[{{google books |plainurl=y |id=4U-5BQAAQBAJ|page=748 }} 748]}}</ref> This amounts to the further constraint that the convolution of {{mvar|η<sub>ε</sub>}} with {{mvar|η<sub>δ</sub>}} must satisfy
<math display="block">\eta_\varepsilon * \eta_\delta = \eta_{\varepsilon+\delta}</math>

for all {{math|1=''ε'', ''δ'' > 0}}. Convolution semigroups in {{math|''L''<sup>1</sup>}} that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction.

In practice, semigroups approximating the delta function arise as [[Fundamental solution|fundamental solution]]s or [[Green's function]]s to physically motivated [[Elliptic partial differential equation|elliptic]] or [[Parabolic partial differential equation|parabolic]] partial differential equations.  In the context of [[Applied mathematics|applied mathematics]], semigroups arise as the output of a [[Linear time-invariant system|linear time-invariant system]].  Abstractly, if ''A'' is a linear operator acting on functions of ''x'', then a convolution semigroup arises by solving the [[Initial value problem|initial value problem]]

<math display="block">\begin{cases}
\dfrac{\partial}{\partial t}\eta(t,x) = A\eta(t,x), \quad t>0 \\[5pt]
\displaystyle\lim_{t\to 0^+} \eta(t,x) = \delta(x)
\end{cases}</math>

in which the limit is as usual understood in the weak sense.  Setting {{math|1=''η<sub>ε</sub>''(''x'') = ''η''(''ε'', ''x'')}} gives the associated nascent delta function.

Some examples of physically important convolution semigroups arising from such a fundamental solution include the following.

====The heat kernel====
The [[Heat kernel|heat kernel]], defined by

<math display="block">\eta_\varepsilon(x) = \frac{1}{\sqrt{2\pi\varepsilon}} \mathrm{e}^{-\frac{x^2}{2\varepsilon}}</math>

represents the temperature in an infinite wire at time {{math|1=''t'' > 0}}, if a unit of heat energy is stored at the origin of the wire at time {{math|1=''t'' = 0}}. This semigroup evolves according to the one-dimensional [[Heat equation|heat equation]]:

<math display="block">\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}.</math>

In [[Probability theory|probability theory]], {{math|1=''η<sub>ε</sub>''(''x'')}} is a [[Normal distribution|normal distribution]] of [[Variance|variance]] {{mvar|ε}} and mean {{math|0}}. It represents the [[Physics:Probability density function|probability density]] at time {{math|1=''t'' = ''ε''}} of the position of a particle starting at the origin following a standard [[Brownian motion]]. In this context, the semigroup condition is then an expression of the [[Markov property]] of Brownian motion.

In higher-dimensional Euclidean space {{math|'''R'''<sup>''n''</sup>}}, the heat kernel is
<math display="block">\eta_\varepsilon = \frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x}{2\varepsilon}},</math>
and has the same physical interpretation, {{lang|la|[[Philosophy:Mutatis mutandis|mutatis mutandis]]}}. It also represents a nascent delta function in the sense that {{math|''η<sub>ε</sub>'' → ''δ''}} in the distribution sense as {{math|''ε'' → 0}}.

====The Poisson kernel====
The [[Poisson kernel]]
<math display="block">\eta_\varepsilon(x) = \frac{1}{\pi}\mathrm{Im}\left\{\frac{1}{x-\mathrm{i}\varepsilon}\right\}=\frac{1}{\pi} \frac{\varepsilon}{\varepsilon^2 + x^2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} \xi x-|\varepsilon \xi|}\,d\xi</math>

is the fundamental solution of the Laplace equation in the upper half-plane.{{sfn|Stein|Weiss|1971|loc=§I.1}}  It represents the electrostatic potential in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the [[Cauchy distribution]] and [[Kernel (statistics)#Kernel functions in common use|Epanechnikov and Gaussian kernel]] functions.<ref>{{Cite book|last=Mader|first=Heidy M.|url={{google books |plainurl=y |id=e5Y_RRPxdyYC}}|title=Statistics in Volcanology|date=2006|publisher=Geological Society of London|isbn=978-1-86239-208-3|language=en|page=[{{google books |plainurl=y |id=e5Y_RRPxdyYC|page=81}} 81]}}</ref> This semigroup evolves according to the equation
<math display="block">\frac{\partial u}{\partial t} = -\left (-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}u(t,x)</math>

where the operator is rigorously defined as the Fourier multiplier
<math display="block">\mathcal{F}\left[\left(-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}f\right](\xi) = |2\pi\xi|\mathcal{F}f(\xi).</math>

===Oscillatory integrals===
In areas of physics such as [[Physics:Wave propagation|wave propagation]] and [[Wave|wave mechanics]], the equations involved are hyperbolic and so may have more singular solutions.  As a result, the nascent delta functions that arise as fundamental solutions of the associated [[Cauchy problem]]s are generally [[Oscillatory integral|oscillatory integral]]s.  An example, which comes from a solution of the [[Euler–Tricomi equation]] of [[Physics:Transonic|transonic]] gas dynamics,{{sfn|Vallée|Soares|2004|loc=§7.2}} is the rescaled [[Airy function]]
<math display="block">\varepsilon^{-1/3}\operatorname{Ai}\left (x\varepsilon^{-1/3} \right). </math>

Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense.  Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the [[Dirichlet kernel]] below), rather than in the sense of measures.

Another example is the Cauchy problem for the [[Physics:Wave equation|wave equation]] in {{math|'''R'''<sup>1+1</sup>}}:{{sfn|Hörmander|1983|loc=§7.8}}
<math display="block"> \begin{align}
c^{-2}\frac{\partial^2u}{\partial t^2} - \Delta u &= 0\\
u=0,\quad \frac{\partial u}{\partial t} = \delta &\qquad \text{for }t=0.
\end{align} </math>

The solution {{mvar|u}} represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin.

Other approximations to the identity of this kind include the [[Sinc function|sinc function]] (used widely in electronics and telecommunications)
<math display="block">\eta_\varepsilon(x)=\frac{1}{\pi x}\sin\left(\frac{x}{\varepsilon}\right)=\frac{1}{2\pi}\int_{-\frac{1}{\varepsilon}}^{\frac{1}{\varepsilon}} \cos(kx)\,dk </math>

and the [[Bessel function]]
<math display="block">  \eta_\varepsilon(x) =  \frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}} \left(\frac{x+1}{\varepsilon}\right). </math>

===Plane wave decomposition===
One approach to the study of a linear partial differential equation
<math display="block">L[u]=f,</math>

where {{mvar|L}} is a [[Differential operator|differential operator]] on {{math|'''R'''<sup>''n''</sup>}}, is to seek first a fundamental solution, which is a solution of the equation
<math display="block">L[u]=\delta.</math>

When {{mvar|L}} is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned).  For more complicated operators, it is sometimes easier first to consider an equation of the form
<math display="block">L[u]=h</math>

where {{mvar|h}} is a [[Physics:Plane wave|plane wave]] function, meaning that it has the form
<math display="block">h = h(x\cdot\xi)</math>

for some vector {{mvar|ξ}}.  Such an equation can be resolved (if the coefficients of {{mvar|L}} are [[Analytic function|analytic function]]s) by the [[Cauchy–Kovalevskaya theorem]] or (if the coefficients of {{mvar|L}} are constant) by quadrature.  So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations.

Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by [[Biography:Johann Radon|Johann Radon]], and then developed in this form by [[Biography:Fritz John|Fritz John]] (1955).{{sfn|Courant|Hilbert|1962|loc=§14}} Choose {{mvar|k}} so that {{math|''n'' + ''k''}} is an even integer, and for a real number {{mvar|s}}, put
<math display="block">g(s) = \operatorname{Re}\left[\frac{-s^k\log(-is)}{k!(2\pi i)^n}\right]
=\begin{cases}
\frac{|s|^k}{4k!(2\pi i)^{n-1}}   &n \text{ odd}\\[5pt]
-\frac{|s|^k\log|s|}{k!(2\pi i)^n}&n \text{ even.}
\end{cases}</math>

Then {{mvar|δ}} is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure {{mvar|dω}} of {{math|''g''(''x'' · ''ξ'')}} for {{mvar|ξ}} in the [[Unit sphere|unit sphere]] {{math|''S''<sup>''n''−1</sup>}}:
<math display="block">\delta(x) = \Delta_x^{(n+k)/2} \int_{S^{n-1}} g(x\cdot\xi)\,d\omega_\xi.</math>

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function {{mvar|φ}},
<math display="block">\varphi(x) = \int_{\mathbf{R}^n}\varphi(y)\,dy\,\Delta_x^{\frac{n+k}{2}} \int_{S^{n-1}} g((x-y)\cdot\xi)\,d\omega_\xi.</math>

The result follows from the formula for the [[Newtonian potential]] (the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the [[Radon transform]] because it recovers the value of {{math|''φ''(''x'')}} from its integrals over hyperplanes.  For instance, if {{mvar|n}} is odd and {{math|1=''k'' = 1}}, then the integral on the right hand side is
<math display="block">
\begin{align}
& c_n \Delta^{\frac{n+1}{2}}_x\iint_{S^{n-1}} \varphi(y)|(y-x) \cdot \xi| \, d\omega_\xi \, dy \\[5pt]
& \qquad = c_n \Delta^{(n+1)/2}_x \int_{S^{n-1}} \, d\omega_\xi \int_{-\infty}^\infty |p| R\varphi(\xi,p+x\cdot\xi)\,dp
\end{align}
</math>

where {{math|''Rφ''(''ξ'', ''p'')}} is the Radon transform of {{mvar|φ}}:
<math display="block">R\varphi(\xi,p) = \int_{x\cdot\xi=p} f(x)\,d^{n-1}x.</math>

An alternative equivalent expression of the plane wave decomposition is:{{sfn|Gelfand|Shilov|1966–1968|loc=I, §3.10}}
<math display="block">\delta(x) = \begin{cases}
  \frac{(n-1)!}{(2\pi i)^n}\displaystyle\int_{S^{n-1}}(x\cdot\xi)^{-n} \, d\omega_\xi & n\text{ even} \\
  \frac{1}{2(2\pi i)^{n-1}}\displaystyle\int_{S^{n-1}}\delta^{(n-1)}(x\cdot\xi)\,d\omega_\xi & n\text{ odd}.
  \end{cases}</math>

===Fourier kernels===
In the study of [[Fourier series]], a major question consists of determining whether and in what sense the Fourier series associated with a [[Periodic function|periodic function]] converges to the function.  The {{mvar|n}}-th partial sum of the Fourier series of a function {{mvar|f}} of period {{math|2π}} is defined by convolution (on the interval {{closed-closed|−π,π}}) with the [[Dirichlet kernel]]:
<math display="block">D_N(x) = \sum_{n=-N}^N e^{inx} = \frac{\sin\left(\left(N+\frac12\right)x\right)}{\sin(x/2)}.</math>
Thus,
<math display="block">s_N(f)(x) = D_N*f(x) = \sum_{n=-N}^N a_n e^{inx}</math>
where
<math display="block">a_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(y)e^{-iny}\,dy.</math>
A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval&nbsp;{{closed-closed|−π,π}} tends to a multiple of the delta function as {{math|''N'' → ∞}}.  This is interpreted in the distribution sense, that
<math display="block">s_N(f)(0) = \int_{-\pi}^{\pi} D_N(x)f(x)\,dx \to 2\pi f(0)</math>
for every compactly supported {{em|smooth}} function {{mvar|f}}.  Thus, formally one has
<math display="block">\delta(x) = \frac1{2\pi} \sum_{n=-\infty}^\infty e^{inx}</math>
on the interval {{closed-closed|−π,π}}.

Despite this, the result does not hold for all compactly supported {{em|continuous}} functions: that is {{math|''D<sub>N</sub>''}} does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of summability methods to produce convergence.  The method of [[Cesàro summation]] leads to the [[Fejér kernel]]{{sfn|Lang|1997|p=312}}

<math display="block">F_N(x) = \frac1N\sum_{n=0}^{N-1} D_n(x) = \frac{1}{N}\left(\frac{\sin \frac{Nx}{2}}{\sin \frac{x}{2}}\right)^2.</math>

The [[Fejér kernel]]s tend to the delta function in a stronger sense that<ref>In the terminology of {{harvtxt|Lang|1997}}, the Fejér kernel is a Dirac sequence, whereas the Dirichlet kernel is not.</ref>

<math display="block">\int_{-\pi}^{\pi} F_N(x)f(x)\,dx \to 2\pi f(0)</math>

for every compactly supported {{em|continuous}} function {{mvar|f}}.  The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point.

===Hilbert space theory===
The Dirac delta distribution is a densely defined [[Unbounded operator|unbounded]] linear functional on the [[Hilbert space]] [[Lp space|L<sup>2</sup>]] of [[Square-integrable function|square-integrable function]]s.  Indeed, smooth compactly supported functions are [[Dense set|dense]] in {{math|''L''<sup>2</sup>}}, and the action of the delta distribution on such functions is well-defined.  In many applications, it is possible to identify subspaces of {{math|''L''<sup>2</sup>}} and to give a stronger [[Topology|topology]] on which the delta function defines a bounded linear functional.

====Sobolev spaces====
The Sobolev embedding theorem for [[Sobolev space]]s on the real line {{math|'''R'''}} implies that any square-integrable function {{mvar|f}} such that

<math display="block">\|f\|_{H^1}^2 = \int_{-\infty}^\infty |\widehat{f}(\xi)|^2 (1+|\xi|^2)\,d\xi < \infty</math>

is automatically continuous, and satisfies in particular

<math display="block">\delta[f]=|f(0)| < C \|f\|_{H^1}.</math>

Thus {{mvar|δ}} is a bounded linear functional on the Sobolev space {{math|''H''<sup>1</sup>}}.  Equivalently {{mvar|δ}} is an element of the continuous dual space {{math|''H''<sup>−1</sup>}} of {{math|''H''<sup>1</sup>}}. More generally, in {{mvar|n}} dimensions, one has {{math|''δ'' ∈ ''H''<sup>−''s''</sup>('''R'''<sup>''n''</sup>)}} provided {{math|''s'' > {{sfrac|''n''|2}}}}.

====Spaces of holomorphic functions====
In [[Complex analysis|complex analysis]], the delta function enters via [[Cauchy's integral formula]], which asserts that if {{mvar|D}} is a domain in the [[Complex plane|complex plane]] with smooth boundary, then

<math display="block">f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z},\quad z\in D</math>

for all [[Holomorphic function|holomorphic function]]s {{mvar|f}} in {{mvar|D}} that are continuous on the closure of {{mvar|D}}.  As a result, the delta function {{math|''δ''<sub>''z''</sub>}} is represented in this class of holomorphic functions by the Cauchy integral:

<math display="block">\delta_z[f] = f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z}.</math>

Moreover, let {{math|''H''<sup>2</sup>(∂''D'')}} be the [[Hardy space]] consisting of the closure in {{math|''L''<sup>2</sup>(∂''D'')}} of all holomorphic functions in {{mvar|D}} continuous up to the boundary of {{mvar|D}}.  Then functions in {{math|''H''<sup>2</sup>(∂''D'')}} uniquely extend to holomorphic functions in {{mvar|D}}, and the Cauchy integral formula continues to hold.  In particular for {{math|''z'' ∈ ''D''}}, the delta function {{mvar|δ<sub>z</sub>}} is a continuous linear functional on {{math|''H''<sup>2</sup>(∂''D'')}}.  This is a special case of the situation in [[Several complex variables|several complex variables]] in which, for smooth domains {{mvar|D}}, the [[Szegő kernel]] plays the role of the Cauchy integral.{{sfn|Hazewinkel|1995|p=[{{google books |plainurl=y |id=PE1a-EIG22kC|page=357}} 357]}}

====Resolutions of the identity====
Given a complete [[Orthonormal basis|orthonormal basis]] set of functions {{math|{{brace|''φ''<sub>''n''</sub>}}}} in a separable Hilbert space, for example, the normalized eigenvectors of a [[Compact operator on Hilbert space#Spectral theorem|compact self-adjoint operator]], any vector {{mvar|f}} can be expressed as
<math display="block">f = \sum_{n=1}^\infty \alpha_n \varphi_n. </math>
The coefficients {α<sub>n</sub>} are found as
<math display="block">\alpha_n = \langle \varphi_n, f \rangle,</math>
which may be represented by the notation:
<math display="block">\alpha_n =  \varphi_n^\dagger f, </math>
a form of the [[Bra–ket notation|bra–ket notation]] of Dirac.<ref>

The development of this section in bra–ket notation is found in {{harv|Levin|2002|loc= Coordinate-space wave functions and completeness, pp.=109''ff''}}</ref> Adopting this notation, the expansion of {{mvar|f}} takes the dyadic form:{{sfn|Davis|Thomson|2000|loc=Perfect operators, p.344}}

<math display="block">f =  \sum_{n=1}^\infty \varphi_n \left ( \varphi_n^\dagger f \right). </math>

Letting {{mvar|I}} denote the identity operator on the Hilbert space, the expression

<math display="block">I = \sum_{n=1}^\infty \varphi_n \varphi_n^\dagger, </math>

is called a [[Borel functional calculus#Resolution of the identity|resolution of the identity]]. When the Hilbert space is the space {{math|''L''<sup>2</sup>(''D'')}} of square-integrable functions on a domain {{mvar|D}}, the quantity:

<math display="block">\varphi_n \varphi_n^\dagger, </math>

is an integral operator, and the expression for {{mvar|f}} can be rewritten

<math display="block">f(x) = \sum_{n=1}^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi) \, d \xi.</math>

The right-hand side converges to {{mvar|f}} in the {{math|''L''<sup>2</sup>}} sense.  It need not hold in a pointwise sense, even when {{mvar|f}} is a continuous function.  Nevertheless, it is common to abuse notation and write

<math display="block">f(x) = \int \, \delta(x-\xi) f (\xi)\, d\xi, </math>

resulting in the representation of the delta function:{{sfn|Davis|Thomson|2000|loc=Equation 8.9.11, p. 344}}

<math display="block">\delta(x-\xi) = \sum_{n=1}^\infty  \varphi_n (x) \varphi_n^*(\xi). </math>

With a suitable [[Rigged Hilbert space|rigged Hilbert space]] {{math|(Φ, ''L''<sup>2</sup>(''D''), Φ*)}} where {{math|Φ ⊂ ''L''<sup>2</sup>(''D'')}} contains all compactly supported smooth functions, this summation may converge in {{math|Φ*}}, depending on the properties of the basis {{math|''φ''<sub>''n''</sub>}}.  In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the [[Distribution (mathematics)#Distributions|distribution]] sense.{{sfn|de la Madrid|Bohm|Gadella|2002}}

===Infinitesimal delta functions===
Cauchy used an infinitesimal {{mvar|α}} to write down a unit impulse, infinitely tall and narrow Dirac-type delta function {{mvar|δ<sub>α</sub>}} satisfying <math display="inline">\int F(x)\delta_\alpha(x) \,dx = F(0)</math> in a number of articles in 1827.{{sfn|Laugwitz|1989}} Cauchy defined an infinitesimal in ''[[Cours d'Analyse]]'' (1827) in terms of a sequence tending to zero.  Namely, such a null sequence becomes an infinitesimal in Cauchy's and [[Biography:Lazare Carnot|Lazare Carnot]]'s terminology.

[[Non-standard analysis]] allows one to rigorously treat infinitesimals. The article by {{harvtxt|Yamashita|2007}} contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the [[Hyperreal number|hyperreals]].  Here the Dirac delta can be given by an actual function, having the property that for every real function {{mvar|F}} one has <math display="inline">\int F(x)\delta_\alpha(x) \, dx = F(0)</math> as anticipated by Fourier and Cauchy.

==Dirac comb==
{{Main|Dirac comb}}
[[File:Dirac comb.svg|thumb|A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of {{mvar|T}}]]
A so-called uniform "pulse train" of Dirac delta measures, which is known as a [[Dirac comb]], or as the Sha distribution, creates a [[Sampling (signal processing)|sampling]] function, often used in [[Digital signal processing|digital signal processing]] (DSP) and discrete time signal analysis.  The Dirac comb is given as the infinite sum, whose limit is understood in the distribution sense,

<math display="block">\operatorname{\text{Ш}}(x) = \sum_{n=-\infty}^\infty \delta(x-n),</math>

which is a sequence of point masses at each of the integers.

Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform.  This is significant because if {{mvar|f}} is any [[Schwartz space|Schwartz function]], then the [[Wrapped distribution|periodization]] of {{mvar|f}} is given by the convolution
<math display="block">(f * \operatorname{\text{Ш}})(x) = \sum_{n=-\infty}^\infty f(x-n).</math>
In particular,
<math display="block">(f*\operatorname{\text{Ш}})^\wedge = \widehat{f}\widehat{\operatorname{\text{Ш}}} = \widehat{f}\operatorname{\text{Ш}}</math>
is precisely the [[Poisson summation formula]].{{sfn|Córdoba|1988}}{{sfn|Hörmander|1983|loc=[{{google books |plainurl=y |id=aaLrCAAAQBAJ|page=177}} §7.2]}}
More generally, this formula remains to be true if {{mvar|f}} is a tempered distribution of rapid descent or, equivalently, if <math>\widehat{f}</math> is a slowly growing, ordinary function within the space of tempered distributions.

==Sokhotski–Plemelj theorem==
The [[Sokhotski–Plemelj theorem]], important in quantum mechanics, relates the delta function to the distribution {{math|p.v. {{sfrac|''x''}}}}, the [[Cauchy principal value]] of the function {{math|{{sfrac|''x''}}}}, defined by

<math display="block">\left\langle\operatorname{p.v.}\frac{1}{x}, \varphi\right\rangle = \lim_{\varepsilon\to 0^+}\int_{|x|>\varepsilon} \frac{\varphi(x)}{x}\,dx.</math>

Sokhotsky's formula states that{{sfn|Vladimirov|1971|loc=§5.7}}

<math display="block">\lim_{\varepsilon\to 0^+} \frac{1}{x\pm i\varepsilon} = \operatorname{p.v.}\frac{1}{x} \mp i\pi\delta(x),</math>

Here the limit is understood in the distribution sense, that for all compactly supported smooth functions {{mvar|f}},

<math display="block">\int_{-\infty}^{\infty}\lim_{\varepsilon\to0^{+}}\frac{f(x)}{x\pm i\varepsilon}\,dx=\mp i\pi f(0)+\lim_{\varepsilon\to0^{+}}\int_{|x|>\varepsilon}\frac{f(x)}{x}\,dx.</math>

==Relationship to the Kronecker delta==
The [[Kronecker delta]] {{mvar|δ<sub>ij</sub>}} is the quantity defined by

<math display="block">\delta_{ij} = \begin{cases} 1 & i=j\\ 0 &i\not=j \end{cases} </math>

for all integers {{mvar|i}}, {{mvar|j}}.  This function then satisfies the following analog of the sifting property: if {{mvar|a<sub>i</sub>}} (for {{mvar|i}} in the set of all integers) is any doubly infinite sequence, then

<math display="block">\sum_{i=-\infty}^\infty a_i \delta_{ik}=a_k.</math>

Similarly, for any real or complex valued continuous function {{mvar|f}} on {{math|'''R'''}}, the Dirac delta satisfies the sifting property

<math display="block">\int_{-\infty}^\infty f(x)\delta(x-x_0)\,dx=f(x_0).</math>

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.{{sfn|Hartmann|1997|loc=pp. 154–155}}

==Applications==

===Probability theory===
In [[Probability theory|probability theory]] and [[Statistics|statistics]], the Dirac delta function is often used to represent a discrete distribution, or a partially discrete, partially continuous distribution, using a [[Physics:Probability density function|probability density function]] (which is normally used to represent absolutely continuous distributions).  For example, the probability density function {{math|''f''(''x'')}} of a discrete distribution consisting of points {{math|1='''x''' = {{brace|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}}}, with corresponding probabilities {{math|''p''<sub>1</sub>, ..., ''p<sub>n</sub>''}}, can be written as

<math display="block">f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

As another example, consider a distribution in which 6/10 of the time returns a standard [[Normal distribution|normal distribution]], and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete mixture distribution).  The density function of this distribution can be written as

<math display="block">f(x) = 0.6 \, \frac {1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} + 0.4 \, \delta(x-3.5).</math>

The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If {{math|1=''Y'' = g(''X'')}} is a continuous differentiable function, then the density of {{mvar|Y}} can be written as

<math display="block">f_Y(y) = \int_{-\infty}^{+\infty} f_X(x) \delta(y-g(x)) \,dx.  </math>

The delta function is also used in a completely different way to represent the [[Local time (mathematics)|local time]] of a [[Diffusion process|diffusion process]] (like [[Brownian motion]]).  The local time of a stochastic process {{math|''B''(''t'')}} is given by
<math display="block">\ell(x,t) = \int_0^t \delta(x-B(s))\,ds</math>
and represents the amount of time that the process spends at the point {{mvar|x}} in the range of the process.  More precisely, in one dimension this integral can be written
<math display="block">\ell(x,t) = \lim_{\varepsilon\to 0^+}\frac{1}{2\varepsilon}\int_0^t \mathbf{1}_{[x-\varepsilon,x+\varepsilon]}(B(s))\,ds</math>
where <math>\mathbf{1}_{[x-\varepsilon,x+\varepsilon]}</math> is the [[Indicator function|indicator function]] of the interval <math>[x-\varepsilon,x+\varepsilon].</math>

===Quantum mechanics===
The delta function is expedient in [[Physics:Quantum mechanics|quantum mechanics]]. The [[Wave function|wave function]] of a particle gives the [[Physics:Probability amplitude|probability amplitude]] of finding a particle within a given region of space.  Wave functions are assumed to be elements of the Hilbert space {{math|''L''<sup>2</sup>}} of [[Square-integrable function|square-integrable function]]s, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set {{math|{{brace|{{ket|''φ<sub>n</sub>''}}}}}} of wave functions is orthonormal if they are normalized by

<math display="block">\langle\varphi_n \mid \varphi_m\rangle = \delta_{nm}</math>

where {{mvar|δ}} is the Kronecker delta.  A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function {{math|{{ket|ψ}}}} can be expressed as a linear combination of the {{math|{{brace|{{ket|''φ<sub>n</sub>''}}}}}} with complex coefficients:

<math display="block"> \psi = \sum c_n \varphi_n, </math>

with {{math|1=''c<sub>n</sub>'' = {{bra-ket|''φ<sub>n</sub>''|''ψ''}}}}.  Complete orthonormal systems of wave functions appear naturally as the [[Eigenfunction|eigenfunction]]s of the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]] (of a [[Physics:Bound state|bound system]]) in quantum mechanics that measures the energy levels, which are called the eigenvalues.  The set of eigenvalues, in this case, is known as the [[Spectrum|spectrum]] of the Hamiltonian.  In [[Bra–ket notation|bra–ket notation]], as above, this equality implies the resolution of the identity:

<math display="block">I = \sum |\varphi_n\rangle\langle\varphi_n|.</math>

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an [[Physics:Observable|observable]] may be continuous rather than discrete.  An example is the [[Physics:Position operator|position observable]], {{math|1=''Qψ''(''x'') = ''x''ψ(''x'')}}.  The spectrum of the position (in one dimension) is the entire real line and is called a [[Physics:Continuous spectrum|continuous spectrum]].  However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions.  The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well: that is, to replace the Hilbert space of quantum mechanics with an appropriate [[Rigged Hilbert space|rigged Hilbert space]].{{sfn|Isham|1995|loc=§6.2}}  In this context, the position operator has a complete set of eigen-distributions, labeled by the points {{mvar|y}} of the real line, given by

<math display="block">\varphi_y(x) = \delta(x-y).</math>

The eigenfunctions of position are denoted by {{math|1=''φ<sub>y</sub>'' = {{ket|''y''}}}} in Dirac notation, and are known as position eigenstates.

Similar considerations apply to the eigenstates of the [[Physics:Momentum operator|momentum operator]], or indeed any other self-adjoint [[Unbounded operator|unbounded operator]] {{mvar|P}} on the Hilbert space, provided the spectrum of {{mvar|P}} is continuous and there are no degenerate eigenvalues.  In that case, there is a set {{math|Ω}} of real numbers (the spectrum), and a collection {{mvar|φ<sub>y</sub>}} of distributions indexed by the elements of {{math|Ω}}, such that

<math display="block">P\varphi_y = y\varphi_y.</math>

That is, {{mvar|φ<sub>y</sub>}} are the eigenvectors of {{mvar|P}}.  If the eigenvectors are normalized so that

<math display="block">\langle \varphi_y,\varphi_{y'}\rangle = \delta(y-y')</math>

in the distribution sense, then for any test function {{mvar|ψ}},

<math display="block"> \psi(x) = \int_\Omega  c(y) \varphi_y(x) \, dy</math>

where {{math|1= ''c''(''y'') = {{angbr|''ψ'', ''φ<sub>y</sub>''}}}}. That is, as in the discrete case, there is a resolution of the identity

<math display="block">I = \int_\Omega |\varphi_y\rangle\, \langle\varphi_y|\,dy</math>

where the operator-valued integral is again understood in the weak sense.  If the spectrum of {{mvar|P}} has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum {{em|and}} an integral over the continuous spectrum.

The delta function also has many more specialized applications in quantum mechanics, such as the [[Physics:Delta potential|delta potential]] models for a single and double potential well.

===Structural mechanics===
The delta function can be used in [[Physics:Structural mechanics|structural mechanics]] to describe transient loads or point loads acting on structures. The governing equation of a simple [[Physics:Harmonic oscillator|mass–spring system]] excited by a sudden force [[Physics:Impulse|impulse]] {{mvar|I}} at time {{math|1=''t'' = 0}} can be written

<math display="block">m \frac{d^2 \xi}{dt^2} + k \xi = I \delta(t),</math>

where {{mvar|m}} is the mass, {{mvar|ξ}} is the deflection, and {{mvar|k}} is the spring constant.

As another example, the equation governing the static deflection of a slender [[Physics:Beam (structure)|beam]] is, according to Euler–Bernoulli theory,

<math display="block">EI \frac{d^4 w}{dx^4} = q(x),</math>

where {{mvar|EI}} is the [[Bending stiffness|bending stiffness]] of the beam, {{mvar|w}} is the [[Engineering:Deflection (engineering)|deflection]], {{mvar|x}} is the spatial coordinate, and {{math|''q''(''x'')}} is the load distribution. If a beam is loaded by a point force {{mvar|F}} at {{math|1=''x'' = ''x''<sub>0</sub>}}, the load distribution is written

<math display="block">q(x) = F \delta(x-x_0).</math>

As the integration of the delta function results in the [[Heaviside step function]], it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise [[Polynomial|polynomial]]s.

Also, a point [[Physics:Bending moment|moment]] acting on a beam can be described by delta functions. Consider two opposing point forces {{mvar|F}} at a distance {{mvar|d}} apart. They then produce a moment {{math|1=''M'' = ''Fd''}} acting on the beam. Now, let the distance {{mvar|d}} approach the [[Limit of a function|limit]] zero, while {{mvar|M}} is kept constant. The load distribution, assuming a clockwise moment acting at {{math|1=''x'' = 0}}, is written

<math display="block">\begin{align}
q(x) &= \lim_{d \to 0} \Big( F \delta(x) - F \delta(x-d) \Big) \\[4pt]
&= \lim_{d \to 0} \left( \frac{M}{d} \delta(x) - \frac{M}{d} \delta(x-d) \right) \\[4pt]
&= M \lim_{d \to 0} \frac{\delta(x) - \delta(x - d)}{d}\\[4pt]
&= M \delta'(x).
\end{align}</math>

Point moments can thus be represented by the [[Derivative|derivative]] of the delta function. Integration of the beam equation again results in piecewise [[Polynomial|polynomial]] deflection.

==See also==
*[[Atom (measure theory)]]
*[[Laplacian of the indicator]]

==Notes==
{{clear}}
{{Reflist}}

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==External links==
*{{springer|title=Delta-function|id=p/d030950}}
*[http://www.khanacademy.org/video/dirac-delta-function KhanAcademy.org video lesson]
*[http://www.physicsforums.com/showthread.php?t=73447 The Dirac Delta function], a tutorial on the Dirac delta function.
*[http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-23-use-with-impulse-inputs Video Lectures – Lecture 23], a lecture by Arthur Mattuck.
*[http://www.osaka-kyoiku.ac.jp/~ashino/pdf/chinaproceedings.pdf The Dirac delta measure is a hyperfunction]
*[http://www.ing-mat.udec.cl/~rodolfo/Papers/BGR-3.pdf We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure]
*[http://www.mathematik.uni-muenchen.de/~lerdos/WS04/FA/content.html Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure.] {{Webarchive|url=https://web.archive.org/web/20080307221128/http://www.mathematik.uni-muenchen.de/~lerdos/WS04/FA/content.html |date=2008-03-07 }}

{{ProbDistributions|miscellaneous}}
{{Differential equations topics}}


[[Category:Fourier analysis]]
[[Category:Generalized functions]]
[[Category:Measure theory]]
[[Category:Digital signal processing]]

{{Sourceattribution|Dirac delta function}}