Dirac delta function in reciprocal function

In summary: A}{x} = A\frac{dlnx}{dx}+Ai\pi\delta(x). This proves that if A=B, then \frac{A}{x}=\frac{B}{x}.Another argument for this formula is by integrating equation (1) from -a to a and assuming in a small region [-\varepsilon, \varepsilon], \int_{-\varepsilon}^{\varepsilon}\frac{1}{x}dx=0. However, in this case, \int_{-a}^{a}c\delta(x)dx=c, which shows that the left side of the equation does not equal the right side. This highlights the importance
  • #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

Thank you for your interesting question. The formula you have mentioned, \frac{A}{x}=\frac{B}{x}+c\delta(x), is derived from the Dirac delta function, which is a mathematical construct used to represent point-like sources or impulses in physical systems.

To understand how this formula is derived, we first need to understand the properties of the Dirac delta function. The Dirac delta function is defined as:

\delta(x) = \begin{cases} +\infty & \text{if } x = 0\\ 0 & \text{if } x \neq 0 \end{cases}

and it satisfies the following properties:

1. \int_{-\infty}^{\infty}\delta(x)dx = 1
2. \delta(x) = 0 \text{ for } x \neq 0
3. For any smooth function f(x), \int_{-\infty}^{\infty}\delta(x)f(x)dx = f(0)

Now, let's look at the formula \frac{dlnx}{dx} = \frac{1}{x}-i\pi\delta(x). This formula is derived from the properties of the Dirac delta function. We can write \frac{dlnx}{dx} as:

\frac{dlnx}{dx} = \frac{d}{dx}ln(x) = \frac{1}{x}

Using the second property of the Dirac delta function, we can write \frac{d}{dx}ln(x) = \frac{1}{x}-i\pi\delta(x). This formula is known as the sifting property of the Dirac delta function.

Now, let's go back to the formula \frac{A}{x}=\frac{B}{x}+c\delta(x). This formula is derived by using the sifting property of the Dirac delta function. We can write \frac{A}{x} and \frac{B}{x} as:

\frac{A}{x} = A\frac{dlnx}{dx}+Ai\pi\delta(x)
\frac{B}{x} = B\frac{dlnx}{dx}+Bi\pi\delta(x)

Since A=B, we can substitute B for A in the second equation, giving us:

 

1. What is the Dirac delta function?

The Dirac delta function, also known as the Dirac delta distribution, is a mathematical function that is used to represent an infinitely sharp peak at a specific point on a number line. It is commonly used in physics and engineering to model impulse or spike-like phenomena.

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

The Dirac delta function can be defined in terms of the reciprocal function as the limit of a sequence of functions, where each function is a scaled and shifted version of the reciprocal function. This relationship allows for the use of the Dirac delta function in solving problems involving the reciprocal function.

3. What is the integral of the Dirac delta function?

The integral of the Dirac delta function is equal to 1. This is because the Dirac delta function has an infinite height at the point where it is defined, but has a width of 0. Therefore, when integrated over an interval containing the point where it is defined, the integral will be equal to the area of the infinitely narrow rectangle, which is equal to 1.

4. Can the Dirac delta function be graphically represented?

Technically, the Dirac delta function cannot be graphed as it is not a traditional function with a defined graph. However, it can be represented graphically as an infinitely tall, infinitely narrow spike at the point where it is defined, with a total area of 1 under the curve. This graphical representation is often used in physics and engineering to visualize the behavior of the Dirac delta function.

5. How is the Dirac delta function used in real-world applications?

The Dirac delta function is used in a variety of fields, including physics, engineering, and signal processing. It is particularly useful in modeling and solving problems involving impulse or spike-like phenomena, such as particle interactions, electrical circuits, and signal filtering. It can also be used in probability and statistics to represent random variables with discrete distributions.

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