Kronecker delta and Dirac delta

In summary, the conversation discusses the validity of the statement \frac{\delta _{n}^{x} }{h} \rightarrow \delta (x-n)as h tends to 0, with the first function being the Kronecker delta and the second being the Dirac delta. One person suspects it is true but cannot provide a proof. Another person argues that it is not true in a meaningful sense and discusses the convergence of generalized distributions. The conversation also touches on the relationship between Kronecker and Dirac's delta.
  • #1
Klaus_Hoffmann
86
1
I do not know if it is true but is this identity true

[tex] \frac{\delta _{n}^{x} }{h} \rightarrow \delta (x-n) [/tex]

as h tends to 0 ?, the first is Kronecker delta the second Dirac delta.

i suspect that the above it is true but can not give a proof
 
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  • #2
It isn't true in any meaningful sense - of course, since you've not defined what you mean by convergent, then we're going to have to guess. What does convergence of generalized distributions even mean? The most meaningful, and traditional notion of a sequence of functions converging to the Dirac delta is as follows:

we have functions f_r(x) for r in the natural numbers, normally, we require the integral of f_r over the real line to be 1 for all r, and for the sequence of numbers

[tex]y_r:=\int f_r(x)g(x)dx[/tex]

to converge to g(0).

These are all trivially false in your case. It is true for suitably normalized Gaussians, for instance.
 
  • #3
ok, thanks i thought it was true by the similarity of the results

[tex] \sum_{n=0}^{\infty} f(n) \delta _{n}^{2} =f(2) [/tex]

(due to Kronecker delta only the value f(2) is obtained in the end)

and [tex] \int_{0}^{\infty} f(x) \delta (x-2)dx =f(2) [/tex]

in the first case f(n) takes only discrete values, whereas in the second f(x) involves a sum over the interval (0, infinity) although the sum and the series give the same result f(2)
 
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  • #4

What is the Kronecker delta?

The Kronecker delta, denoted by δ, is a mathematical function that takes two arguments, typically integers, and evaluates to 1 if the arguments are equal and 0 otherwise. It is often used in linear algebra and discrete mathematics.

What is the Dirac delta?

The Dirac delta, denoted by δ, is a generalized function in mathematics that represents a point mass or impulse at a specific point. It is frequently used in physics and engineering to describe phenomena such as point charges and gravitational forces.

What are the properties of the Kronecker delta?

The Kronecker delta has several properties, including symmetry, transitivity, and reflexivity. It also follows the Kronecker delta identity, which states that δ(i,j)δ(j,k) = δ(i,k).

How is the Dirac delta used in integration?

The Dirac delta is often used in integration as a mathematical tool to represent a discrete set of points. It can be used to simplify integrals and solve differential equations, and it is closely related to the concept of the unit impulse function in engineering.

What are the relationships between the Kronecker delta and Dirac delta?

The Kronecker delta and Dirac delta are closely related mathematical concepts. The Kronecker delta can be seen as a discrete version of the Dirac delta, where the latter is a continuous function. The Dirac delta can also be expressed in terms of the Kronecker delta in certain situations, such as when integrating over a discrete set of points.

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