Dirac delta function in reciprocal function

In summary, the conversation discusses Dirac's formula that states if A=B, then \frac{A}{x}=\frac{B}{x}+c\delta(x) and how it is derived. It is shown that if A=B, one can't infer A/x=B/x, but only A/x=B/x+cδ(x). The conversation also mentions the identity \lim_{\epsilon \rightarrow 0} \frac{1}{x\pm i\epsilon} = \mathcal P \frac{1}{x} \mp i\pi \delta(x), but it is unclear how it relates to the formula.
  • #1
jackychenp
28
0
From dirac, if A=B, then [itex] \frac{A}{x}=\frac{B}{x}+c\delta(x)[/itex] (1) How this formula is derived?

Since [itex]\frac{dlnx}{dx} = \frac{1}{x}-i\pi\delta(x)[/itex]
We can get [itex]\frac{A}{x} = A\frac{dlnx}{dx}+Ai\pi\delta(x)[/itex]
[itex]\frac{B}{x} = B\frac{dlnx}{dx}+Bi\pi\delta(x)[/itex]
So if A=B, [itex] \frac{A}{x}=\frac{B}{x}.[/itex]

Another argument is if we integrate the equation (1) from -a to a, a->[itex]\infty[/itex] and assume in a small region [itex][-\varepsilon, \varepsilon ][/itex], [itex]\int_{-\varepsilon}^{\varepsilon}\frac{1}{x}dx=0,[/itex] so we can get [itex] \int_{-a}^{a}\frac{1}{x}dx=0, but \int_{-a}^{a}c\delta(x)dx=c,[/itex] so the left side of equation (1) doesn't equal to the right side. Please correct me if I am wrong!
 
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  • #2
I'm not sure what you're getting at. It seems like you're talking about the identity

[tex]\lim_{\epsilon \rightarrow 0} \frac{1}{x\pm i\epsilon} = \mathcal P \frac{1}{x} \mp i\pi \delta(x),[/tex]
where [itex]\mathcal P[/itex] denotes a principle value.

Beyond that, though, I'm not sure what your question is.
 
  • #3
Hi Mute,

In Dirac's book, he demonstrates that if A=B, one can't infer A/x=B/x, but only A/x=B/x+cδ(x). I cannot get the latter result. And if we integrate the latter equation, it doesn't look correct.

Mute said:
I'm not sure what you're getting at. It seems like you're talking about the identity

[tex]\lim_{\epsilon \rightarrow 0} \frac{1}{x\pm i\epsilon} = \mathcal P \frac{1}{x} \mp i\pi \delta(x),[/tex]
where [itex]\mathcal P[/itex] denotes a principle value.

Beyond that, though, I'm not sure what your question is.
 

1. What is the Dirac delta function?

The Dirac delta function, denoted by δ(x), is a mathematical function that is defined to be zero everywhere except at x=0, where it is infinite. It is often used in physics and engineering to represent impulsive forces or point charges.

2. How is the Dirac delta function related to the reciprocal function?

The Dirac delta function is related to the reciprocal function, 1/x, through the following property: ∫-∞ δ(x) / x dx = 1. This means that the integral of the Dirac delta function divided by x is equal to 1.

3. Can the Dirac delta function be graphed?

Technically, the Dirac delta function cannot be graphed since it is infinite at x=0. However, it is often represented by a spike at x=0 with an area of 1, which can help visualize its properties.

4. How is the Dirac delta function used in mathematics and science?

The Dirac delta function is used in many areas of mathematics and science, including differential equations, signal processing, and quantum mechanics. It is particularly useful for representing point sources or impulses in these fields.

5. Are there any limitations to using the Dirac delta function?

One limitation of the Dirac delta function is that it is not a true function in the traditional sense, as it is undefined at x=0. Additionally, its use can lead to nonsensical results if not used correctly, so care must be taken when manipulating equations involving the Dirac delta function.

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