Exploring the Properties of the Dirac Delta Function

In summary, the conversation is discussing how to prove the statement: \int_a^b f(x)g' (x)\, dx = -f(0). The speaker suggests using integration by parts, but is unsure of how to proceed after that. They also mention some properties related to the Dirac Delta Function, but are unsure of how it relates to the statement. They ask for a tip or additional constraints that may help with the proof.
  • #1
arierreF
79
0
Prove that.

[itex]\int_a^b f(x)g' (x)\, dx = -f(0)[/itex]


This is supposed to be a delta Dirac function property. But i can not prove it.
I thought using integration by parts.

[itex] \int_a^b f(x)g' (x)\, dx = f(x)g(x) - \int_a^b f(x)'g (x)\, dx [/itex]

But what now?


Some properties:


[itex] \delta [g(x)] = \sum \frac{1}{|g'(xi)|} [/itex]

[itex] \int_a^b f(x)\delta(x-xi)\, dx = [/itex]

[itex]f(x_{0})[/itex] if [itex]a<x_{0}<b[/itex]
0, other cases.




I just need a tip please.
 
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  • #2
arierreF said:
Prove that.

[itex]\int_a^b f(x)g' (x)\, dx = -f(0)[/itex]
In general, this is wrong. Are there any additional constraints on f,g,a,b?
If that would be true, all integrals would be trivial ;).
 
  • #3
What does this have to do with the "Dirac Delta Function"? Is g' supposed to be the Dirac Delta Function? What is g?
 

Related to Exploring the Properties of the Dirac Delta Function

What is the Delta Dirac function property?

The Delta Dirac function property, also known as the Dirac delta function, is a mathematical function that is often used in physics and engineering to represent a point mass or impulse. It is defined as a function that is zero everywhere except at one point, where it is infinite, and its integral over the entire real line is equal to one.

How is the Delta Dirac function used in applications?

The Delta Dirac function is used to model point sources or impulses in various physical systems, such as in signal processing, control theory, and quantum mechanics. It is also used in Fourier analysis to represent the spectrum of signals and in probability theory to represent probability distributions.

What are the properties of the Delta Dirac function?

The Delta Dirac function has several important properties, including the sifting property, which states that the integral of the function over a set containing the point where it is non-zero is equal to the value of the function at that point. It also has a scaling property, which allows it to be scaled by a constant factor without affecting its integral value. Additionally, it has a symmetry property and a time-shifting property.

How is the Delta Dirac function related to the Kronecker delta?

The Delta Dirac function is a continuous version of the Kronecker delta, which is a discrete function that takes on the value of 1 at one point and 0 everywhere else. They both have similar properties, with the main difference being that the Delta Dirac function is a continuous function while the Kronecker delta is a discrete function.

What are some common notations for the Delta Dirac function?

The Delta Dirac function is often denoted using various notations, including δ(x), δ(x - a), δ(t), or δ[n], depending on the context and the variable being used. It is also sometimes written as δa(x) to indicate the point where it is non-zero.

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