Dirac Delta Function Properties

[Emc2brain]

Okay...so here's the thing. I have been researching the dirac Delta properties. The sights I've visited, thus far, are moderately helpful. I'm looking to tackle this question I'm about to propose, so for you Brains out there (the truly remarkable ) please don't post a solution, pointers in the right direction would be really great.

I'm in the process of showing that the following two distinct property (nascent) are in actuality valid...


nascent property I:
pie*delta(y) = lim(n-->infinity) of sin(n*y)/y

nascent property II:
2*pie*delta(y) = integral[-infinity-->+infinity] e^(iky)dk



Hannah

 

[Allday]

Hi Hannah,

The delta function is a wonderful way of approximating many things in physics and was thankfully given a rigorous grounding by mathematicians.

The first property is one (of the many) series of functions that converges to the delta function as n goes to infinity. I would look in mathematical methods for physics books for discussions of this.

The second property is something very usefull that comes out of Fourier analysis. Any good coverage of Fourier will discuss that one.

This might be a little too broad, but I didn't want to give anything away for you. BTW are you trying to rigorously prove those properties or just show that they are plausible?

 

[Emc2brain]

Yes, rigorous. I know what they are and have used their applications, just having some difficulty in proving it mathematically...

Thanx
Hannah

 

[HallsofIvy]

Part of the problem is that the Dirac Delta "function" is not a function! It is, rather, a "generalized function" or "distribution". Schwarz wrote "the book" on them in terms of functionals. That is, a "distribution" is an operator that, to every function (in a certain function space) assigns a number. In particular, the Dirac Delta function assigns to the function f(x) the number f(0).