Proof involving Dirac Delta function

In summary, the conversation discussed problem 1.45 from Griffiths' book, which is to prove that x * d/dx [δ(x)] = -δ(x). Different approaches were attempted, including using integration by parts and taking the derivative of both sides, but neither proved successful. The second part of the problem involved proving that the derivative of the step function, θ(x), is equal to δ(x). It was suggested to integrate against a test function, f(x), and show that integrating f(x) * x * δ'(x) is the same as integrating f(x) * (-δ(x)). This method involves introducing a dummy function, which may seem strange but can lead to the correct answer.
  • #1
122
0
Prove that
[itex] x \frac{d}{dx} [\delta (x)] = -\delta (x) [/itex]
this is problem 1.45 out of griffiths book by the way.

Homework Equations


I attempted to use integration by parts as suggest by griffiths using [itex] f = x , g' = \frac{d}{dx}[/itex]
This yields [itex] x [\delta (x)] - \int \delta (x)dx[/itex]

next I tried taking the derivative of both sides so that I would get my original
[itex] x \frac{d}{dx} [\delta (x)] = x \frac{d}{dx}[\delta (x)] [/itex]

and so I'm back where I started. I have also tried using a definite integral so that

[itex]\int _{-\infty }^{\infty }x \frac{d}{dx}[\delta (x)] \text{dx} = \int_{-\infty }^{\infty } x[\delta (x)] \, dx - \int_{-\infty }^{\infty } [\delta (x)] \, dx [/itex]

but we know that [itex]\int_{-\infty }^{\infty } x[\delta (x)] \, dx = 0 [/itex] so our equation simplifies.
However this didn't get me any closer to solving the problem either.

I also have the second part of the problem regarding the step function which is defined as
[itex] \theta (x)
\begin{array}{ll}
\{ &
\begin{array}{ll}
1 & x>0 \\
0 & x\leq 0 \\
\end{array}
\\
\end{array}
[/itex] Show that [itex] \frac{d}{dx}\theta (x) =\delta (x)[/itex]

This something I can grasp by stating that

[itex]
\frac{\Delta \theta }{\text{$\Delta $x}}(x=0 )\longrightarrow 1/0, \text{ as } \text{$\Delta $x}\longrightarrow 0.[/itex]
which is clearly an undefined(or infinite) slope, and at every other point on this graph the derivative is going to be 0. However I want to know if this is sufficiently rigorous, and if it isn't what a step in the right direction might be.
 
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  • #2
For the first part you should integrate against a test function f(x). Show that integrating f(x)*x*δ'(x) is the same as integrating f(x)*(-δ(x)).
 
  • #3
Alright thanks I will try that. Can you give some sort of reasoning for why this method works to get the answer? It just seems strange to me to introduce a dummy function.
 

1. What is the Dirac Delta function?

The Dirac Delta function, denoted as δ(x), is a mathematical concept used in the field of calculus and signal processing. It is defined as a function with a value of 0 everywhere except at x = 0, where its value is considered to be infinite. It is often used to represent a point of concentration or impulse in a mathematical model.

2. What are some common applications of the Dirac Delta function?

The Dirac Delta function has various applications in physics, engineering, and mathematics. Some common uses include modeling point charges in electrostatics, representing the impulse response of a system in signal processing, and solving differential equations with discontinuous functions.

3. How is the Dirac Delta function related to the Kronecker Delta function?

The Dirac Delta function and the Kronecker Delta function are both special types of mathematical functions that have values of 0 everywhere except at a specific point. However, the Dirac Delta function is a continuous function, while the Kronecker Delta function is a discrete function. The Dirac Delta function is also defined in terms of an integral, while the Kronecker Delta function is defined as a sum.

4. Can the Dirac Delta function be integrated?

Yes, the Dirac Delta function can be integrated, but it is not a regular function and cannot be integrated in the traditional sense. Instead, it is commonly integrated using the concept of the Riemann-Stieltjes integral, which is a generalization of the standard Riemann integral.

5. How is the Dirac Delta function used in Fourier analysis?

In Fourier analysis, the Dirac Delta function is used to represent a single frequency component in a signal. It allows us to decompose a complex signal into its individual frequency components and study their contributions. The Dirac Delta function is also used in the Fourier Transform, where it acts as a weighting function to represent the strength of each frequency component.

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