Book to study Dirac delta function from a physics point of view [duplicate]

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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.

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

• 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. – Zach466920 Apr 4 '16 at 19:16
• $\int F(x)\delta(x-a)dx=F(a)$ is the only other property I ever needed to know beyond the two listed. – Kyle Kanos Apr 4 '16 at 19:27