Where can I find the proof of Dirac's function properties?

In summary, The delta function, also known as the Dirac delta function, is a distribution that is defined by its property of mapping a function to its value at 0 when integrated. It is not an ordinary function and is usually studied in the field of distribution theory in calculus and analysis. To understand its representation as a limit, one may refer to a tutorial provided in the conversation.
  • #1
abc def
1
0
Where can i find the proof of dirac's function properties?
 
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  • #2
The delta "function" isn't an ordinary function; it's a distribution, and it only makes sense when you integrate it. It's *defined* by the property
[tex]\int_{-\infty}^\infty \delta(x) f(x) \;dx = f(0)[/tex]
for suitable test functions [itex]f[/itex].
 
  • #3
In other words, the delta function is the operator that maps f(x) to f(0).

This doesn't belong in this area. I am moving it to "Calculus and Analysis"
 
  • #4
abc def said:
Where can i find the proof of dirac's function properties?

Look for a textbook that sounds like "Distribution theory" and you'll find what you need.
 
  • #5
abc def said:
Where can i find the proof of dirac's function properties?
If you wish to understand what the Dirac delta "really is", and how it might be represented as a sort of "limit", then you may read the following tutorial:
https://www.physicsforums.com/showthread.php?t=73447
 
  • #6
arilno, that link is invalid.
 
  • #7
I've fixed it now.
 

1. What is the Dirac (delta) function?

The Dirac (delta) function, also known as the Dirac delta distribution, is a mathematical function that is defined as zero everywhere except at the origin, where it is infinite. It is often used in physics and engineering to model point-like or concentrated forces or impulses.

2. How is the Dirac (delta) function represented mathematically?

The Dirac (delta) function is represented mathematically as δ(x) or δ0(x), where x is the independent variable. It can also be written as a limit of a sequence of functions, such as the Gaussian function or the rectangular function.

3. What are the properties of the Dirac (delta) function?

The Dirac (delta) function has several important properties, including:

  • δ(x) = 0 for all x ≠ 0
  • ∫δ(x)dx = 1
  • ∫f(x)δ(x)dx = f(0) (the sifting property)
  • δ(ax) = δ(x)/|a| for any non-zero constant a
  • δ(x) = δ(-x) (symmetry property)

4. How is the Dirac (delta) function used in physics?

The Dirac (delta) function is used in physics to model point-like or localized forces or impulses. For example, it is used to represent the force exerted by a particle on a mass, or the voltage spike caused by a sudden change in current. It is also used in quantum mechanics to describe the position and momentum of particles.

5. Can the Dirac (delta) function be graphed?

Technically, the Dirac (delta) function cannot be graphed since it is an infinitely thin function. However, it can be visualized as a tall and narrow spike at the origin, with a height of infinity. Alternatively, it can be approximated by a very narrow and tall rectangular pulse, which becomes narrower and taller as the pulse's area approaches 1.

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