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Area Mathematics - Functions / Distributions

IEV ref103-03-05

Symbol
δ

en
Dirac function
Dirac delta function
unit pulse
unit impulse, US
distribution assigning to any function f(x), continuous for x=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaaiaadIhacqGH9aqpju gqbiaaicdaaaa@38BA@ , the value f(0)

Note 1 to entry: The Dirac function can be considered as the limit of a function, equal to zero outside a small interval containing the origin, and the integral of which remains equal to unity when this interval tends to zero. See Figure 2, where instead of a triangle any other shape with area 1 is possible, too.

Note 2 to entry: The Dirac function is the derivative of the unit step function considered as a distribution.

Note 3 to entry: The Dirac function can be defined for any value x0 of the variable x. The usual notation is:

f( x 0 )= + δ(x x 0 )f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaam iEamaaBaaaleaajugWaiaaicdaaSqabaGccaGGPaGaeyypa0Zaa8qm aeaaimaacqWF0oazcaGGOaGaamiEaiabgkHiTiaadIhadaWgaaWcba qcLbmacaaIWaaaleqaaaqaaiaayIW7cqGHsislcqGHEisPaeaacaaM i8Uaey4kaSIaeyOhIukaniabgUIiYdGccaGGPaGaamOzaiaacIcaca WG4bGaaiykaKqzaeGaciizaOGaamiEaaaa@5364@

Figure 1 – Distribution de Dirac

Figure 1 – Dirac function


fr
distribution de Dirac, f
impulsion unité, f
percussion unité, f
distribution associant à toute fonction f(x), continue pour x=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaaiaadIhacqGH9aqpju gqbiaaicdaaaa@38BA@ , la valeur f(0)

Note 1 à l'article: La distribution de Dirac peut être considérée comme la limite d'une fonction nulle en dehors d'un petit intervalle contenant l'origine et dont l'intégrale reste égale à l'unité lorsque cet intervalle tend vers zéro. Voir la Figure 2, où le triangle peut être remplacé par n’importe quelle forme d’aire unité.

Note 2 à l'article: La distribution de Dirac est la dérivée de la fonction échelon unité considérée comme une distribution.

Note 3 à l'article: La distribution de Dirac peut être définie pour toute valeur x0 de la variable x. La notation usuelle est:

f( x 0 )= + δ(x x 0 )f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWaamGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaam iEamaaBaaaleaajugWaiaaicdaaSqabaGccaGGPaGaeyypa0Zaa8qm aeaaimaacqWF0oazcaGGOaGaamiEaiabgkHiTiaadIhadaWgaaWcba qcLbmacaaIWaaaleqaaaqaaiaayIW7cqGHsislcqGHEisPaeaacaaM i8Uaey4kaSIaeyOhIukaniabgUIiYdGccaGGPaGaamOzaiaacIcaca WG4bGaaiykaKqzaeGaciizaOGaamiEaaaa@5364@


ar
دالة ديراك
دالة ديراك التركيبية
النبضة الواحدة
قفزة الوحدة

de
Dirac-Funktion, f
Delta-Distribution, f
Einheitsstoßfunktion, f

es
distribución de Dirac

it
distribuzione di Dirac
funzione delta di Dirac
impulso unitario

ja
ディラック関数
ディラックのデルタ関数
単位パルス
単位インパルス

pl
dystrybucja Diraca
delta Diraca
impuls jednostkowy
funkcja Diraca (termin niezalecany)

pt
função de Dirac
impulso unitário

sr
Диракова функција, ж јд
јединична импулсна функција, ж јд

sv
Diracs deltafunktion
Diracs funktion
enhetspuls

zh
狄拉克函数
狄拉克δ 函数
单位脉冲函数
单位冲激函数

Publication date: 2009-12
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