- #1
Swapnil
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https://www.physicsforums.com/showthread.php?t=73447
I saw the above tutorial by arildno and looked at how he defined the Dirac Delta "function" as a functional. But isn't there a more easier way to do this. I have seen the following definition in a lot of textbooks.
[tex]\delta(t) \triangleq \lim_{\epsilon \to 0} \frac{1}{\epsilon} \Pi\Big(\frac{t}{\epsilon}\Big)[/tex]
where [tex]\Pi(t)[/tex] is the gate function and is defined as
[tex]
\Pi (t) :=
\begin{cases}
0 & \mbox{ for } |x| > \frac{1}{2} \\
\frac{1}{2} & \mbox{ for } |x| = \frac{1}{2} \\
1 & \mbox{ for } |x| < \frac{1}{2},
\end{cases}
[/tex]
What's wrong by defining the delta function in this way?
I saw the above tutorial by arildno and looked at how he defined the Dirac Delta "function" as a functional. But isn't there a more easier way to do this. I have seen the following definition in a lot of textbooks.
[tex]\delta(t) \triangleq \lim_{\epsilon \to 0} \frac{1}{\epsilon} \Pi\Big(\frac{t}{\epsilon}\Big)[/tex]
where [tex]\Pi(t)[/tex] is the gate function and is defined as
[tex]
\Pi (t) :=
\begin{cases}
0 & \mbox{ for } |x| > \frac{1}{2} \\
\frac{1}{2} & \mbox{ for } |x| = \frac{1}{2} \\
1 & \mbox{ for } |x| < \frac{1}{2},
\end{cases}
[/tex]
What's wrong by defining the delta function in this way?
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