Mathematical help in understanding this integral (Dirac formulation)

In summary, the Dirac delta function is a generalized function with a Fourier transform equal to 1. It is not a function in the traditional sense, but rather a sequence of functions that converge to a common limit. This concept was developed to address issues with limits and non-differentiable functions. There are multiple spaces of generalized functions and further study is recommended for a deeper understanding.
  • #1
PhyAmateur
105
2
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
 
Physics news on Phys.org
  • #2
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".
 
  • #3
Will do, thanks.
 

1. What is the purpose of using the Dirac formulation in mathematical integrals?

The Dirac formulation is a mathematical tool used to simplify the representation of integrals involving complex functions. It allows for the integration of functions that are not normally integrable, making calculations easier and more efficient.

2. How does the Dirac formulation differ from traditional integral methods?

The Dirac formulation differs from traditional integral methods in that it uses the concept of delta functions to represent the integrand. This allows for the integration of functions that are not continuous or do not have a definite value at certain points.

3. Can the Dirac formulation be used for all types of integrals?

No, the Dirac formulation is mainly used for integrals involving highly oscillatory or singular functions. It is not suitable for all types of integrals and should only be used when traditional methods are not applicable.

4. Is there a specific technique for solving integrals using the Dirac formulation?

Yes, the most commonly used technique for solving integrals using the Dirac formulation is the method of steepest descent. This involves finding the saddle points of the integrand and using them to approximate the integral.

5. Are there any limitations or drawbacks to using the Dirac formulation?

Yes, the Dirac formulation can only be used for integrals involving functions that are not too complex or rapidly oscillating. It also requires a good understanding of complex analysis and may not be suitable for beginners in mathematics.

Similar threads

Replies
1
Views
833
  • Calculus
Replies
25
Views
912
Replies
2
Views
845
Replies
32
Views
3K
  • Calculus
Replies
4
Views
1K
Replies
3
Views
661
Replies
16
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
227
Replies
5
Views
1K
Replies
3
Views
929
Back
Top