Most general form of nascent delta function

In summary, The speaker is looking for a general formula for all nascent delta functions, as they have only found random forms. They are interested in a function or functional g that satisfies the condition lim x -> 0 g(x,y) = delta(y) and are struggling to find a linear combination of these functions. They have tried using Fourier series and transform, but have not been successful. They have discovered that what they are looking for is a linear combination of probability distributions that converge to the Dirac delta. However, there is no information about the number of distributions that satisfy this condition. The speaker concludes that the Dirac delta is not special and there is no classification for things that converge to it other than "it converges to the
  • #1
mmzaj
107
0
Dear all

I'm wondering if you can help me find the most general formula of all nascent delta functions. all i have found a somewhat random forms . I'm looking for a general elegant formula that all the forms can be derived from .

thanks in advance .
 
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  • #3
thanks a lot , really informative , but here is the thing , I'm looking for a function ( or a functional ) g such that :

lim x ->0 g (x,y) = delta(y).

well , obviously any nascent delta function satisfies the condition , but the problem is that I am also interested in g(x,y) every where , not only when x -> 0 , and there is no constrains on g(x,y) other than it equals delta when x->0 . hence , the program is to find the most general form of the nascent delta functions , and construct a linear combination of them . i tried Fourier series and transform to check if there are common properties of the nascent delta functions in the frequency domain - as we electrical engineers like to call it - but in vain . here is the dilemma . can you help me here .

thanks in advance .
 
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  • #4
now , after a long struggle i found that what I'm looking for isn't any thing but a linear combination of probability distributions ( such as the normal distribution ) G (x,y) such that lim x->0 G(x,y) = delta (y) . but again , there is no information about the number of distributions that satisfy that condition . the normal distribution does , cauchy distribution does , but are there other distributions like that ?
 
  • #5
The Dirac delta is not special; it's just another point in the space of distributions. There are exactly as many things that converge to the zero distribution as there are that converge to the Dirac delta distribution.

There really isn't a classification for things that converge to the Dirac delta other than "it converges to the Dirac delta".
 

1. What is a nascent delta function?

A nascent delta function is a mathematical concept that represents a very narrow, infinitely tall spike or impulse at a specific point on a graph. It is often used in physics and engineering to approximate an ideal point source or to describe the behavior of a system near a singularity.

2. What is the general form of a nascent delta function?

The general form of a nascent delta function is a Dirac delta function, denoted by δ(x), which is equal to zero for all values of x except at x = 0 where it is equal to infinity. This can also be written as δ(x-a), where a is the location of the spike on the graph.

3. How is a nascent delta function different from a regular delta function?

A regular delta function, also known as a Kronecker delta function, is equal to 1 when the argument is zero, and 0 otherwise. It is a discrete version of the Dirac delta function, which is a continuous function. The nascent delta function is a more general form of the Dirac delta function that can be used to represent a wider range of phenomena.

4. What are some real-world applications of nascent delta functions?

Nascent delta functions have many applications in physics, particularly in quantum mechanics, where they are used to describe the behavior of particles in a confined space. They are also used in signal processing to represent ideal impulse signals, and in engineering to model point sources of energy or forces in a system.

5. How is the concept of a nascent delta function related to limits and derivatives?

The concept of a nascent delta function is closely related to limits and derivatives, as it can be thought of as the limiting case of a function with a very narrow width and infinitely large height. In fact, the Dirac delta function can be defined as the derivative of the Heaviside step function, and can also be used to evaluate certain integrals using the concept of the limit.

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