The nature of the dirac delta function

In summary, the conversation discusses the use of the dirac delta function in an equation involving 1/x. The definition of the dirac delta function is explained and it is mentioned that while it may be useful to visualize it as a function, its actual definition is more complex. The use of the dirac delta function in the equation is found to be problematic and it is suggested to use the properties commonly attributed to the function instead. Ultimately, the equation can be simplified to 1/x = 1/x, but it is noted that this equation has no meaning at x=0.
  • #1
sgfw
15
0
From what I can tell, it seems that 1/x + δ(x) = 1/x because if we think of both 1/x and the dirac delta function as the following peicewise functions:

1/x = 1/x for x < 0
1/x = undefined for x = 0
1/x = 1/x for x > 0

δ(x) = 0 for x < 0
δ(x) = undefined for x = 0
δ(x) = 0 for x > 0

then for any part of 1/x that is defined, 0 is being added to it. Is it okay to ignore the fact that the function is being changed at x = 0 because it was already undefined?
 
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  • #2
Hi,
This won't work, because your definition of the dirac delta "function" is heuristical, and if you "rigorize" it, your equation won't work (if you actually to try to make it rigorous, you'll face another problem, namely that 1/x is not locally integrable, and you'll have to look at what is called the principal value of 1/x, but you can replace 1/x by 1/x², your argument remains the same, in the proper context, you'll see that it does not work).
 
  • #3
I don't understand-Am I thinking of the dirac delta function in the wrong way? could you explain?
 
  • #4
The dirac delta function is not a function. It is often presented as the function satisfying f(x)=0 for x non zero, f(0)=∞, and such that ∫f=1.
Of course no such function exists, and its actual definition is more sophisticated.
It is not really a bad way to picture that "function" in your head, but it's not usable as such. You have to use the actual definition, or just use the properties commonly attributed to this function (the fact that is is neutral for the convolution, that its Fourier transoform is 1 etc...), all this properties can of course be proved by using the rigorous definition of δ.
 
  • #5
Okay, so if I change my question to "does 1/x + 0/x = 1/x", would the answer be any different?
 
  • #6
Well, 0/x is just the zero function over ℝ-{0}, so... your equation is tantamount to 1/x=1/x over ℝ-{0}, which is of course correct.
 
  • #7
So it doesn't matter that I am adding undefined to undefined at x = 0?
 
  • #8
Your equation has no meaning at x=0.
You have equality of your two sides for all x non zero. There's not much more that can be said.
 
  • #9
okay, thanks.
 

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 used to model a point or spike in a system. It is defined as zero everywhere except at the origin, where it is infinite, and has an integral of one over its domain.

2. What is the physical interpretation of the Dirac Delta Function?

The Dirac Delta Function is often used to represent a point mass or a concentrated force in physics. It can also be used to describe the concentration of charge or mass at a specific point in space, or the density of a fluid at a certain point.

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

The Kronecker Delta Function is a discrete version of the Dirac Delta Function. While the Dirac Delta Function is defined as a continuous function, the Kronecker Delta Function is a discrete function that takes on a value of one at only one point and zero everywhere else. As the spacing between the points approaches zero, the Kronecker Delta Function approaches the Dirac Delta Function.

4. What are some applications of the Dirac Delta Function?

The Dirac Delta Function has many applications in physics, engineering, and mathematics. It is commonly used in signal processing to model a sudden impulse or spike in a signal. It is also used in quantum mechanics to represent a particle confined to a point in space. In electrical engineering, the Dirac Delta Function is used to model an ideal impulse in a circuit.

5. Can the Dirac Delta Function be integrated?

Yes, the Dirac Delta Function can be integrated over its domain. However, it is important to note that the integral of the Dirac Delta Function is not a regular function, but rather a distribution. This means that its integral must be interpreted in the context of a larger function or system, and may not have a numerical value.

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