Mathematical help in understanding this integral (Dirac formulation)

[PhyAmateur]

I was studying Dirac delta function, when I ran across this statement:

The Dirac delta function is defined by: $$\delta(x) = lim _{K \to \infty } d_K(x), $$ where

$$ d_K(x) = \int_{-K/2}^{K/2} \frac{dk}{2\pi} e^{ikx}$$

I was thinking that as $$K\to \infty$$ the integral won't be defined. What's going on here?

Thank you

 

[pwsnafu]

That is certainly not a definition. For a start if you want to define Dirac as sequences of functions you need to define a collection of such sequences, and then consider the common limit. One specific sequence is no where enough. What they are trying to say is that "the Dirac delta has Fourier transform equal to 1".

There is no function whose domain and co-domain are reals, and possessing the properties of Dirac delta. These functions are properly called generalized functions. They were discovered/invented because (1) the paradoxes that arose from limit passage of non-uniform convergence and (2) the desire to differentiate continuous but not differentiable functions. This means your definitions of derivative and integral fail when talking about them.

What makes this confusing is that there are multiple spaces of generalized functions. If you have done linear algebra, you would have done abstract vector spaces. You know there are lots of vectors spaces other than $\mathbb{R}^n$, and the same thing happens here.

This is a very very very large topic, which of course they don't teach you. My recommendation is to go to your library and search for textbook specifically on "generalized functions" or "distribution theory".

 

[PhyAmateur]

Will do, thanks.