Editing Dirac delta function (section)

{{Short description|Generalized function whose value is zero everywhere except at zero}}
{{redirect|Delta function|other uses|Delta function (disambiguation)}}
{{Use American English|date=January 2019}} 
[[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]]s <math>\delta_a(x) = \frac{1}{\left|a\right| \sqrt{\pi}} e^{-(x/a)^2}</math> ]]
{{Differential equations}}

In [[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]] on the [[real numbers]], whose value is zero everywhere except at zero, and whose [[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}} Thus it can be represented heuristically as

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

such that

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

Since there is no function having this property, modelling 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 [[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 [[Laurent Schwartz]] developed the theory of distributions, where it is defined as a linear form acting on functions.