Representation of Delta Function

In summary, the conversation discusses the representation of sin(ax)/x as a delta function in the limit a->infinity. The properties of this representation are stated as 1) it equals 0 unless x=0, 2) integrated from plus minus infinity it equals 1, and 3) multiplying by an arbitrary function f(x) and integrating gives f(0). The conversation also mentions attempts to show these properties, including using series representation and employing residue calculus. However, there is a missing factor of pi in the given representation and the use of residue calculus may not be applicable for all functions. The concept of a test function is also brought up.
  • #1
poonintoon
17
0
Hopefully people are still prowling the forums this close to christmas :)

I want to show that sin(ax)/x is a representation of a delta function in the limit a->infinty i.e
1) It equals 0 unless x=0
2) integrated from plus minus infinity it equals 1 and
3) multiplying by an arbitrary function f(x) and integrating gives f(0)

but I cannot show any of these. I have tried series representation, writing out as exponentials, looking up definite integrals etc but cannot make any headway.

Cheers
 
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  • #2
The integral of your representation can be found by employing residue calculus. I believe you have a factor of [tex]\pi[/tex] missing.
 
  • #3
Hi,

Thanks, you are right about the pi factor. But what if the function f(x) was say x then there would not be a pole and residue calculus wouldn't make sense would it?
 
  • #4
Not every function is allowed as a test function. See

http://en.wikipedia.org/wiki/Distribution_(mathematics [Broken])
 
Last edited by a moderator:

1. What is a delta function?

A delta function, also known as the Dirac delta function, is a mathematical function that is used to represent an infinitely narrow and tall spike at a specific point in space or time. It is often used in physics and engineering to model point-like sources or events.

2. What is the mathematical representation of a delta function?

The mathematical representation of a delta function is written as δ(x-a), where a is the point at which the spike occurs. This function is zero everywhere except at x=a, where it is infinite. However, it is important to note that this is a theoretical concept and not an actual function that can be graphed.

3. How is the delta function used in calculations?

The delta function is often used in integrals to represent a point source or event. It can also be used to simplify complex mathematical expressions and solve differential equations. In physics, it is commonly used in fields such as quantum mechanics and signal processing.

4. What are some properties of the delta function?

Some properties of the delta function include: it is an even function, meaning δ(x) = δ(-x); it is infinitely differentiable at all points except x=0; and it follows the sifting property, meaning that when integrated with a function, it picks out the value of the function at the point where the spike occurs.

5. Are there different types of delta functions?

Yes, there are different types of delta functions, such as the discrete delta function, which is used in discrete systems, and the multidimensional delta function, which is used in multiple dimensions. There are also generalized delta functions, such as the Dirac comb function, which is used to model periodic events.

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