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\begin{align} \delta(x)\delta(x) = ? \end{align}

\begin{align} \delta(x)\delta(x-a) = ? \end{align}

\begin{align} \delta(x)/x = ? \end{align}

In the above problems, a is a real number and a $\ne$ 0, My solutions are below, but I am not sure of them.

\begin{align} \delta(x)\delta(x) = \delta(x=0) \delta(x)=\delta(x) \end{align}

\begin{align} \delta(x)\delta(x-a) = \delta(x=a) \delta(x-a)=0 \end{align}

since the $\delta$(x=a)=0

\begin{align} \delta(x)/x = ? \end{align} I am not sure of this.

Thanks for your feedback.

Adi
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  • About $\delta(x)/x$, see my answer to this recent question : https://math.stackexchange.com/q/3214395 and the first reference therein. – Jean Marie May 20 '19 at 23:30
  • If $a=0$, would your answer to the second one be true? – AnonSubmitter85 May 21 '19 at 00:14
  • @JeanMarie thanks for that, but is the derivative of $\delta$ defined ? – Adi May 21 '19 at 01:57
  • @AnonSubmitter85 thanks for pointing that out what is your opinion of when a not equal to 0. When a=0 , I believe the answer should be just $\delta$(x) ^2. – Adi May 21 '19 at 02:00
  • $\delta(x)^2$ is not defined. – md2perpe May 21 '19 at 06:18
  • @Adi The derivative of $\delta$ is a well defined concept (unlike its square) ; besides, physically, it modelizes a doublet (2 infinitely close opposite charges +q,-q) https://en.wikipedia.org/wiki/Unit_doublet – Jean Marie May 21 '19 at 07:35
  • The function $\delta(x)^2$ may not be defined in a strict mathematical sense, but since you tagged this with 'signal-processing', I'd say that $\delta(x)^2=\delta(x)$ would probably be fine with your professor. Take my advice at your own risk though. – AnonSubmitter85 May 22 '19 at 18:11

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