Properties of the dirac delta function

In summary, the conversation discusses methods to demonstrate the property of differentiating and integrating the delta function. The individual suggests differentiating with respect to x' and using integration by parts, while also questioning if this will shift the argument of f. The other individual mentions Shankar's method on page 62 and agrees that it is a more direct and tidy approach.
  • #1
ptabor
15
0
I'm trying to show that
[tex] \int \delta \prime(x-x')f(x') dx = f\prime(x) [/tex]
can I differentiate delta with respect to x' instead (giving me a minus sign), and then integrate by parts and note that the delta function is zero at the boundaries? this will give me an integral involving f' and delta, so the f' would come out - but I'm not sure that this will shift the argument of f to x.
Shankar demonstrates the property on page 62, but I'd like to know if my method is valid.
 
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  • #2
Hm, assuming that the integration on the left hand side is actually over [itex]x^\prime[/itex] and that [itex]x[/itex] is within the limits of the integration I believe that you can prove this using integration by parts.

Edit: Sorry, that's what you said :smile:

I took a look at Shankar and I like his way better.
 
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  • #3
yes, his derivation is more tidy and more direct. i did the proof using integration by parts first, and then happened upon his.
 

1. What is the Dirac Delta function?

The Dirac Delta function, also known as the Dirac Delta distribution or impulse function, is a mathematical function that is defined to be zero everywhere except at a single point, where it is infinite. It is often used in mathematics and physics to represent a point-like source of energy or mass.

2. What are the properties of the Dirac Delta function?

Some of the key properties of the Dirac Delta function include: it is infinitely tall at its single point of non-zero value, it has a total area of one, it is even, and it has a shifting property that allows it to shift to different locations on the x-axis.

3. How is the Dirac Delta function used in mathematics and physics?

The Dirac Delta function is used in a variety of applications in both mathematics and physics. In mathematics, it is often used to define distributions and as a tool for solving differential equations. In physics, it is commonly used to represent a point charge or mass, or as a way to approximate a point-like phenomenon.

4. What is the relationship between the Dirac Delta function and the Kronecker Delta function?

The Kronecker Delta function is a discrete version of the Dirac Delta function, where it is defined to be 1 at one particular point and 0 everywhere else. The Kronecker Delta function is often used in discrete mathematics, while the Dirac Delta function is used in continuous mathematics and physics.

5. Are there any real-life applications of the Dirac Delta function?

Yes, the Dirac Delta function has many real-life applications in physics and engineering. It is often used in signal processing to represent a short pulse or an impulse response, in quantum mechanics to describe quantum states, and in electrical engineering to model electric potential or current distributions. It also has applications in fluid dynamics and probability theory.

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