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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d0309502.png" />-function, Dirac delta-function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d0309503.png" />''
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''$\delta$-function, Dirac delta-function, $\delta(x)$''
  
A function which makes it possible to describe the spatial density of a physical magnitude (mass, charge, intensity of a heat source, force, etc.) which is concentrated or applied at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d0309504.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d0309505.png" />. For instance, using the delta-function the density of a point mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d0309506.png" /> located at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d0309507.png" /> is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d0309508.png" />. The delta-function may be formally defined by the relation
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A function which makes it possible to describe the spatial density of a physical magnitude (mass, charge, intensity of a heat source, force, etc.) which is concentrated or applied at a point $a$ of a space $\mathbf R^n$. For instance, using the delta-function the density of a point mass $m$ located at a point $a$ is written as $m\delta(x-a)$. The delta-function may be formally defined by the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d0309509.png" /></td> </tr></table>
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\[\int\limits_{\mathbf R^n}\delta(x-a)f(x)\,dx=f(a)\]
  
for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095010.png" />. The derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095011.png" /> of the delta-function may be defined in a similar manner:
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for any continuous function $f$. The derivatives $\delta^{(k)}$ of the delta-function may be defined in a similar manner:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095012.png" /></td> </tr></table>
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\[\int\limits_{\mathbf R^n}\delta^k(x-a)f(x)\,dx=(-1)^kf^{(k)}(a)\]
  
for the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095013.png" /> that are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095014.png" /> with derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095015.png" /> up to the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095016.png" /> inclusive. The formal operator relations, which are frequently employed, and which express the following properties of the delta-function:
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for the class of functions $f$ that are continuous in $\mathbf R^n$ with derivatives $f^{(k)}$ up to the order $k$ inclusive. The formal operator relations, which are frequently employed, and which express the following properties of the delta-function:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095017.png" /></td> </tr></table>
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\[\delta(-x)=\delta(x);\qquad\delta(cx)=|c|^{-1}\delta(x),\quad c=\mathrm{const},\]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095018.png" /></td> </tr></table>
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\[x\delta(x)=0;\qquad\delta(x)+x\delta'(x)=0,\]
  
etc., should be understood in the sense of the above definitions, i.e. these relations become meaningful only after having been integrated against sufficiently smooth functions. Thus, the delta-function is not an ordinary function in the sense of the classical theory of functions, and is defined in the theory of generalized functions as a singular [[Generalized function|generalized function]], i.e. as the continuous linear functional in the space of infinitely-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095019.png" /> of compact support, assigning to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095020.png" /> its value at zero: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095021.png" />.
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etc., should be understood in the sense of the above definitions, i.e. these relations become meaningful only after having been integrated against sufficiently smooth functions. Thus, the delta-function is not an ordinary function in the sense of the classical theory of functions, and is defined in the theory of generalized functions as a singular [[Generalized function|generalized function]], i.e. as the continuous linear functional in the space of infinitely-differentiable functions $f$ of compact support, assigning to $f$ its value at zero: $(\delta,f)=f(0)$.
  
  
  
 
====Comments====
 
====Comments====
The Dirac delta-function is the derivative (in the sense of distributions or generalized functions) of the Heaviside function (Heaviside distribution) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095022.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030950/d03095026.png" /> (the value at zero does not matter; as usual for a distribution it suffices for it to be defined apart from a set of measure zero).
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The Dirac delta-function is the derivative (in the sense of distributions or generalized functions) of the [[Heaviside function]] (Heaviside distribution) $h$, defined by $h(x)=0$ for $x<0$, $h(x)=1$ for $x>0$ (the value at zero does not matter; as usual for a distribution it suffices for it to be defined apart from a set of measure zero).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , '''1–2''' , Hermann  (1966)  {{MR|0209834}} {{ZBL|0149.09501}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.M. Gel'fand,  G.E. Shilov,  "Generalized functions" , '''1. Properties and operations''' , Acad. Press  (1964)  (Translated from Russian)  {{MR|435831}} {{ZBL|0115.33101}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , '''1–2''' , Hermann  (1966)  {{MR|0209834}} {{ZBL|0149.09501}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.M. Gel'fand,  G.E. Shilov,  "Generalized functions" , '''1. Properties and operations''' , Acad. Press  (1964)  (Translated from Russian)  {{MR|435831}} {{ZBL|0115.33101}} </TD></TR></table>

Latest revision as of 22:12, 31 December 2018

$\delta$-function, Dirac delta-function, $\delta(x)$

A function which makes it possible to describe the spatial density of a physical magnitude (mass, charge, intensity of a heat source, force, etc.) which is concentrated or applied at a point $a$ of a space $\mathbf R^n$. For instance, using the delta-function the density of a point mass $m$ located at a point $a$ is written as $m\delta(x-a)$. The delta-function may be formally defined by the relation

\[\int\limits_{\mathbf R^n}\delta(x-a)f(x)\,dx=f(a)\]

for any continuous function $f$. The derivatives $\delta^{(k)}$ of the delta-function may be defined in a similar manner:

\[\int\limits_{\mathbf R^n}\delta^k(x-a)f(x)\,dx=(-1)^kf^{(k)}(a)\]

for the class of functions $f$ that are continuous in $\mathbf R^n$ with derivatives $f^{(k)}$ up to the order $k$ inclusive. The formal operator relations, which are frequently employed, and which express the following properties of the delta-function:

\[\delta(-x)=\delta(x);\qquad\delta(cx)=|c|^{-1}\delta(x),\quad c=\mathrm{const},\]

\[x\delta(x)=0;\qquad\delta(x)+x\delta'(x)=0,\]

etc., should be understood in the sense of the above definitions, i.e. these relations become meaningful only after having been integrated against sufficiently smooth functions. Thus, the delta-function is not an ordinary function in the sense of the classical theory of functions, and is defined in the theory of generalized functions as a singular generalized function, i.e. as the continuous linear functional in the space of infinitely-differentiable functions $f$ of compact support, assigning to $f$ its value at zero: $(\delta,f)=f(0)$.


Comments

The Dirac delta-function is the derivative (in the sense of distributions or generalized functions) of the Heaviside function (Heaviside distribution) $h$, defined by $h(x)=0$ for $x<0$, $h(x)=1$ for $x>0$ (the value at zero does not matter; as usual for a distribution it suffices for it to be defined apart from a set of measure zero).

References

[a1] L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1966) MR0209834 Zbl 0149.09501
[a2] I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1. Properties and operations , Acad. Press (1964) (Translated from Russian) MR435831 Zbl 0115.33101
How to Cite This Entry:
Delta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Delta-function&oldid=28169
This article was adapted from an original article by V.D. Kukin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article