# Definition:Dirac Delta Distribution

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## Definition

Let $a \in \R^d$ be a real vector.

Let $\phi \in \map \DD {\R^d}$ be a test function.

Let $\delta_a \in \map {\DD'} {\R^d}$ be a distribution.

Suppose $\delta_a$ is such that:

- $\forall \phi \in \map \DD {\R^d} : \map {\delta_a} \phi = \map \phi a$

Then $\delta_a$ is known as the **Dirac delta distribution**.

Further research is required in order to fill out the details.In particular: For $d \ge 2$ this works in Euclidean space with Cartesian coordinates. Change of coordinates and integration measure may affect this somewhatYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Research}}` from the code. |

## Source of Name

This entry was named for Paul Adrien Maurice Dirac.

## Also known as

$\delta_a$ is also written as $\map \delta a$.

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples