# What is Dirac delta function?

• I
• Leo Liu
In summary, the conversation discusses the concept of the Dirac delta function and its connection to Heaviside step function. The Dirac delta function is a way to model an instantaneous surge in a distribution, and it is often used in physics to represent momentum. It is not commonly taught at a young age, but it can be understood through the use of functional analysis.

#### Leo Liu

TL;DR Summary
I came across it in the derivation of Gauss' law of electric flux from Coulomb's law. I did some research on it, but the wikipedia page about it was slightly confusing. All I know about it is that it models an instantaneous surge by a distribution. However I am still perplexed by this concept. Can someone please explain it to me like I am a 5-year-old with some calculus or direct me to some useful resources other than wiki?

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NTuft
There are multiple approaches to delta function. Here I would start from
Heaviside Step Funiiton
https://en.wikipedia.org/wiki/Heaviside_step_function
We can not differentiate Heaviside Step Function at 0 in usual healthy sense. But Dirac boldly invented a strange delta "function" as the result of its derivative including x=0.
$$H'(x)=\delta(x)$$

Say we give 1 Ns impulse to a 1kg body to make it move with 1m/s.
There are many ways to do it, e.g.
Applying Force 1 N during 1 second,
Force 10N during 1/10 second,
Force 100N during 1/100second,
-----
Force 1/x N during x second,
------

We can make time duration x as small as we like with increasing Force 1/x .

There is no limit of x to zero in this sequence because 1/x diverges to infinity. But Dirac say there exists force limit ##\delta(x)##, that is to say, in a instant of time, the infinite force is applied to generate 1 Ns momentum on the body. Delta function is a momentum maker in a instant of time.
$$\int_{-\infty}^x \delta(\xi) d\xi = H(x)$$
Heaviside step function shows thus made momentum by Delta function at x=0.

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NTuft, DaveE, atyy and 1 other person
Leo Liu said:

Can someone please explain it to me like I am a 5-year-old ...
The Dirac delta function is rarely taught at infant school. Especially these days.

berkeman and atyy
PeroK said:
The Dirac delta function is rarely taught at infant school. Especially these days.
What's the world coming to these days ?! A B C Dd !

atyy and PeroK
anuttarasammyak said:
There are multiple approaches to delta function. Here I would start from
Heaviside Step Funiiton
https://en.wikipedia.org/wiki/Heaviside_step_function
Thank you. This concept isn't hard to grasp at all. I will look into its connection with Dirac delta.

##\delta##-function is a linear continuous functional on the space of ##C^\infty(\mathbb{R})##-functions with compact support. By definition ##\delta(\varphi)=\varphi(0)##.
The formulas like ##H'=\delta## are understood in the generalized sense
$$-\int_{\mathbb{R}}H(x)\varphi'(x)dx=\varphi(0)$$
For details see any textbook on functional analysis. For example https://www.springer.com/gp/book/9783540586548

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NTuft, atyy and Leo Liu