Proof of Dirac delta sifting property.

In summary, the conversation discusses how to prove a statement involving the Dirac delta function. Suggestions are given, including using the result for f(0) and using a different kind of limit. It is noted that the definition of the Dirac delta in the linked PDF may have a mistake or omission, and a relevant property of this type of limit is mentioned.
  • #1
Lavabug
866
37

Homework Statement


Prove the statement
http://www.mathhelpforum.com/math-help/vlatex/pics/60_32c8daf48ffa5f233ecc2ac3660e517e.png [Broken]


The Attempt at a Solution


I am clueless as to how I would go about doing this, I know the basic properties. I think it has to do with using epsilon somewhere and taking the limit as epsilon approaches zero, as shown here:
http://www-thphys.physics.ox.ac.uk/people/JohnMagorrian/mm/dirac.pdf [Broken]
but I really have no idea how they're using it. The prof did something similar in class but he used -epsilon to epsilon in the limits of integration to show that the integral of δ(x)f(x) is just f(0).

Pretty mysterious to me, any help is greatly appreciated.
 
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  • #2
Lavabug said:

Homework Statement


Prove the statement
http://www.mathhelpforum.com/math-help/vlatex/pics/60_32c8daf48ffa5f233ecc2ac3660e517e.png [Broken]


The Attempt at a Solution


I am clueless as to how I would go about doing this, I know the basic properties. I think it has to do with using epsilon somewhere and taking the limit as epsilon approaches zero, as shown here:
http://www-thphys.physics.ox.ac.uk/people/JohnMagorrian/mm/dirac.pdf [Broken]
but I really have no idea how they're using it. The prof did something similar in class but he used -epsilon to epsilon in the limits of integration to show that the integral of δ(x)f(x) is just f(0).

Pretty mysterious to me, any help is greatly appreciated.

Two suggestions you might try.

1. If you have the result for f(0) try letting u = t-a in this problem. Or

2. Parrot your prof's proof only using an integral from a-ε to a+ε.
 
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  • #3
In your class, how is the dirac delta defined?


The PDF you linked makes a mistake in its definition of the dirac delta, or more accurately a (rather common) omission -- the limit isn't a limit of functions as you learned in calculus class. It's a different sort of limit, whose relevant property is that if [itex]\varphi[/itex] is a test function, then
[tex]\int_{-\infty}^{+\infty} \left( \lim_{\epsilon \to 0}^{\wedge} f_\epsilon(x) \right) \varphi(x) \, dx = \lim_{\epsilon \to 0} \int_{-\infty}^{+\infty} f_\epsilon(x) \varphi(x) \, dx[/tex]
(the limit with the hat is the new kind of limit, the other limit is the ordinary kind you learn in calculus)
(and, for the record, the integral on the left is not the same integral you learned in calculus either)
 

1. What is "Proof of Dirac delta sifting property"?

"Proof of Dirac delta sifting property" is a mathematical concept that refers to the behavior of the Dirac delta function, which is a mathematical tool commonly used in physics and engineering to model point sources. The sifting property states that when the delta function is integrated with a continuous function, the result is equal to the value of the function at the point where the delta function is centered.

2. Why is the Dirac delta function important?

The Dirac delta function is important because it allows us to mathematically represent and analyze point sources, such as particles, in physical systems. It is also used in signal processing to simplify calculations and in quantum mechanics to describe the state of a particle at a specific location.

3. How is the Dirac delta sifting property proven?

The Dirac delta sifting property can be proven using mathematical techniques, such as the definition of the delta function and the fundamental theorem of calculus. By manipulating the equations and applying mathematical rules, it can be shown that the sifting property holds true.

4. What are some real-world applications of the Dirac delta sifting property?

The Dirac delta sifting property has many applications in physics and engineering. It is used in the analysis of electric circuits, in the study of fluid dynamics, in quantum mechanics, and in signal processing. It also plays a role in solving differential equations and other mathematical problems.

5. Are there any limitations to the Dirac delta sifting property?

While the Dirac delta sifting property is a useful mathematical tool, it does have some limitations. It is only valid for continuous functions, and it can only be used in one-dimensional systems. In addition, the delta function is not a true function, so it cannot be graphed or manipulated using traditional algebraic methods.

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