Functional analysis Distributions and generalized functions Object

Dirac delta function

Builds on Heaviside step function Laplace transform

Description

Let $C_c^\infty(\mathbb{R})$ be the space of infinitely differentiable functions $\varphi : \mathbb{R} \to \mathbb{C}$ that vanish outside a bounded set. The Dirac delta is the distribution $\delta$ that maps each test function $\varphi \in C_c^\infty(\mathbb{R})$ to $\delta[\varphi] = \varphi(0)$ . Informally, one writes $\int_{-\infty}^{\infty} \delta(t)\, \varphi(t)\, dt = \varphi(0)$ , as if $\delta$ were a function that is $0$ away from $0$ and has total integral $1$ . No actual function has this property. The delta is a generalized function.

Remarks

As a measure, $\delta$ gives the mass $1$ to each set that contains $0$ and the mass $0$ to the other sets. The shifted delta $\delta(t - a)$ maps $\varphi$ to $\varphi(a)$ . For $a > 0$ , its Laplace transform is $e^{-a s}$ , and for $a = 0$ the usual convention gives $1$ . In the sense of distributions, $\delta$ is the derivative of the Heaviside step function. In differential equations, $\delta(t - a)$ models an impulse: a large force that acts for a very short time around $t = a$ .

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Tag	`0KPHENZ5S7`
URI	`m2://0KPHENZ5S7`
Aliases	`unit-impulse`, `dirac-delta`
Kind	object
Writing	W1 stated
MSC	46F10
Uses	Heaviside step function, Laplace transform
Same as	Wikidata Q209675 Wikipedia Dirac delta function

Dirac delta function

Description

Remarks

Reading path

Where this leads

Further reading

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