Representations of a delta function

[Frabjous]

There are many representations of the delta function. Is there a place/reference that lists AND proves them? I am interested in proofs that would satisfy a physicist not a mathematician.

 

[marcusl]

By representations, do you mean the generating functions that produce the delta in the limit? Physics references do not generally provide proofs so much as demonstrations or explanations, so if you want a proof you should stick to math texts. The list of texts containing demonstrations/discussions, on the other hand, is nearly limitless. Open up your favorite one. Here are a few examples

• Arfken, Mathematical Methods for Physicists
• Bracewell, The Fourier Transform and Its Applications
• Dirac, The Principles of Quantum Mechanics (of course!)

 

[BvU]

representations of the delta function. Is there a place/reference that lists AND proves them

I wonder what you would consider a proof of a representation . A delta function is a something with certain well described properties. The word distributions comes to mind (out of the dust of ages). Showing that the representation exhibits such properties is just that: a showing.

[Links added after commenting; plenty references for further study. As a physicist I'm comfortable with 'infinitely narrow and high needle with area 1' but I would never claim that would prove anything]

 

[Frabjous]

I am actually comfortable with the delta function. It’s just that I am many years past grad school and was reading something (Intermediate Quantum Mechanics by Bethe) that piqued my interest in how to to show that these representations have the correct properties.

 

[WWGD]

I am actually comfortable with the delta function. It’s just that I am many years past grad school and was reading something (Intermediate Quantum Mechanics by Bethe) that piqued my interest in how to to show that these representations have the correct properties.

What are the properties you refer to?

 

[Frabjous]

That the integral over all space of the representations equals one. I am interested in how to do the integrals. Some of them I can solve, some of them I cannot. I do not want to look them up in an integral table.

 

[S.G. Janssens]

The best way (avoiding magic) to manipulate $\delta$-functions or, generally, distributions, is by first pairing with an arbitrary (i.e. smooth, compactly supported) test function, doing the manipulations (usually involving partial integration) and limits in $\mathbb{R}$ and then, at the end, concluding that a certain identity, or limit representation holds true for the distributions in question, because the test function was chosen arbitrarily.

Any introduction to (or: containing material about) distribution theory will show you this in detail, while keeping the underlying functional analysis out, or to a minimum, so it is accessible to physicists with a good background in multivariable calculus. There is the book by Strichartz and the book by Duistermaat and Kolk, for example.

 

[Frabjous]

I’ll give Strichartz a try. Thanks.