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IEVref: | 103-01-03 | ID: | |

Language: | en | Status: Standard | |

Term: | distribution | ||

Synonym1: | generalized function [Preferred] | ||

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Definition: | continuous linear functional which assigns a real or complex number to any function of a real or complex variable belonging to the class of infinitely differentiable functions vanishing outside a bounded interval or domain Note 1 to entry: A function $D(f)={\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}D(x)f(x)\text{d}x}$ if this integral exists. Note 2 to entry: The derivative of a distribution ${D}^{\prime}(f)=-D(\text{d}f/\text{\hspace{0.05em}}\text{d}x)$. | ||

Publication date: | 2009-12 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00020 (IEV 103) - evaluation. JGO 2017-08-25: Corrected </mtex> to </mtext>. LMO | ||

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Note 1 to entry: A function *D*(*x*) can be considered as a distribution *D* assigning to a function *f*(*x*) the value

$D(f)={\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}D(x)f(x)\text{d}x}$ if this integral exists.

Note 2 to entry: The derivative of a distribution *D* is another distribution *D*′ defined for any function *f*(*x*) by

${D}^{\prime}(f)=-D(\text{d}f/\text{\hspace{0.05em}}\text{d}x)$.