Understanding the Dirac Delta function

  • #1
244
5
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}
 

Answers and Replies

  • #2
Vanadium 50
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Neither of those is correct.
 
  • #3
Ssnow
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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
FactChecker
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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.
 

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