Understanding the Dirac Delta function

In summary, the Dirac delta function is a generalized function that can be defined as a distribution or measure. It is defined by its behavior in an integral, where ∫f(x)δ(x)dx = f(0). The expression ∫δ(x-y)f(x)dx = f(y) is a more accurate representation than the incorrect expressions provided. Additionally, the integral cannot be omitted and the second expression does not make sense.
  • #1
redtree
285
13
I just want to make sure that I am understanding the Dirac Delta function properly. Is the following correct?:

For two variables ##x## and ##y##:
\begin{equation}
\begin{split}
\delta(x-y) f(x) &= f(y)
\end{split}
\end{equation}

And:
\begin{equation}
\begin{split}
\delta(x-x) f(x) &= f(x)
\end{split}
\end{equation}
 
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  • #2
Neither of those is correct.
 
  • #3
The Dirac delta is not a function in the traditional sense, it can be rigorously defined either as a distribution or as a measure.
Ssnow
 
  • #4
The Dirac delta function is defined as a generalized function by its behavior in an integral: ∫f(x)δ(x)dx = f(0).
So in your question, ∫δ(x-y)f(x)dx = f(y) would make better sense than your expression (1). The integral can not be omitted. Your expression (2) does not make sense to me.
 

1. What is the Dirac Delta function?

The Dirac Delta function, denoted as δ(x), is a mathematical function that represents an infinitely tall, infinitely narrow spike at the origin (x = 0) on the real number line. It is also known as the unit impulse function or the unit spike function.

2. What is the purpose of the Dirac Delta function?

The Dirac Delta function is used to model or approximate highly concentrated point masses in mathematical models. It is also used in physics to represent point charges, point masses, and point vortices. In signal processing, it is used to represent an idealized point source in an impulse response function.

3. How is the Dirac Delta function defined mathematically?

The Dirac Delta function is defined as a limit of a sequence of functions. It is represented as δ(x) = limn→∞ fn(x), where fn(x) is the sequence of functions that converge to δ(x). The exact definition may vary depending on the context and application.

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

The Dirac Delta function has several important properties, including:

  • δ(x) = 0 for all x ≠ 0
  • ab δ(x) dx = 1, if a ≤ 0 ≤ b
  • δ(x) = ∞ at x = 0
  • δ(x) = 0 for all x outside the domain of integration
  • δ(x) is an even function, i.e. δ(-x) = δ(x)

5. How is the Dirac Delta function used in practical applications?

The Dirac Delta function is used in a wide range of practical applications in mathematics, physics, and engineering. Some common examples include:

  • In electrical engineering, it is used to model the voltage across a capacitor or the current through an inductor in an electrical circuit.
  • In signal processing, it is used to represent a point source or a delta function impulse in a signal.
  • In quantum mechanics, it is used to represent a point particle or a point charge.
  • In fluid dynamics, it is used to model point vortices or sources in a fluid flow.

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