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This question already has an answer here:

I am a beginning physics graduate student. I am often bewildered by the strange properties of the Dirac delta function such as:

  1. $\delta (a x)= \frac{1}{a} \delta (x)$

  2. The derivative of $\delta (x)$

etc etc.

Such strange properties of $\delta (x)$ are mentioned in the first chapter of Arfken-Weber (7th ed.) without proof. Please suggest me a book from which I can learn the minimum essential mathematics of $\delta (x)$ function that is required to study physics. I am not looking for an advanced mathematical treatment but the book should have the proofs of the theorems stated.

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marked as duplicate by Danu, AccidentalFourierTransform, ACuriousMind, Kyle Kanos, John Rennie Apr 5 '16 at 10:42

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    $\begingroup$ For starters, $\delta(x)$ is considered to be a distribution as opposed to a function, if you're a mathematician that is. I've best understood the 'thing' as a the limit of a sequence of ever narrowing Gaussian distributions. $\endgroup$ – Zach466920 Apr 4 '16 at 19:16
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    $\begingroup$ See e.g. this question. $\endgroup$ – ACuriousMind Apr 4 '16 at 19:17
  • $\begingroup$ $\int F(x)\delta(x-a)dx=F(a)$ is the only other property I ever needed to know beyond the two listed. $\endgroup$ – Kyle Kanos Apr 4 '16 at 19:27
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/127376/2451 $\endgroup$ – Qmechanic Apr 4 '16 at 19:28
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'Mathematical Physics' by Kusse and Westwig is just the thing you need. The fifth chapter is devoted to the Dirac-delta function. The book is fairly easy to understand and provides the proofs of the theorems that are stated in Arfken-Weber.

After having read this, you can read the appendices I and II in Cohen-Tannoudji (Quantum Mechanics) on Fourier transforms and Dirac delta functions respectively. The appendices are in Volume II of the book (the book is a pretty huge one and comes in two volumes).

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