Delta function (hard question)

In summary, the delta function is a mathematical function used to represent an infinitely narrow spike or impulse. It has several important properties and is commonly used in science to model physical phenomena, signal processing, and probability theory. It is closely related to the Kronecker delta and has many real-life applications in physics, engineering, and mathematics.
  • #1
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Homework Statement


Compute ∫[itex]^{∞}_{-∞}[/itex]dx (x2+a2)-1δ(sin(2x)), without calculating the resulting sum.


Homework Equations



This is a very specific integral which ,has a delta function δ operating on sin function

The Attempt at a Solution


Does anyone know this integral ? I haven't seen before any similar examples.
 
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  • #2
You haven't listed all relevant equations yet:
[tex]\delta[f(x)]=\sum_{k} \frac{1}{\left |f'(x_k) \right|} \delta(x-x_k),[/tex]
where [itex]x_k[/itex] runs over all zeros of [itex]f[/itex], which all must be simple zeros of course in order that [itex]f'(x_k) \neq 0[/itex] for all [itex]k[/itex].
 

1. What is the delta function?

The delta function, also known as the Dirac delta function, is a mathematical function that is used to represent an infinitely narrow spike or impulse. It is defined as zero everywhere except at the origin, where it has an infinite value with an integral of one.

2. How is the delta function used in science?

The delta function is commonly used in science to model physical phenomena such as point charges, point masses, and point vortices. It is also used in signal processing and probability theory, among other fields.

3. What are the properties of the delta function?

The delta function has several important properties, including the sifting property (the integral of the delta function over a given interval is equal to the value of the function at the origin), the scaling property (multiplying the argument of the delta function by a constant results in a change in amplitude), and the symmetry property (the delta function is an even function).

4. How is the delta function related to the Kronecker delta?

The delta function is closely related to the Kronecker delta, which is a discrete version of the delta function. The Kronecker delta has a value of one when the two indices are equal and zero otherwise. It can be thought of as a discrete version of the sifting property of the delta function.

5. Are there any real-life applications of the delta function?

Yes, the delta function has many real-life applications. It is used in physics to describe the behavior of particles and fields, in engineering for signal processing, and in mathematics for solving differential equations and modeling phenomena that involve impulses or singularities.

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