What is the Integral of a Dirac-Delta Function with a Sine Argument?

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In summary, the conversation discusses how to integrate the dirac-delta function in the context of solving the equation 0∫2∏ δ(sin θ - k) dθ. The steps involve first understanding the value of the integral when |k| > 1, then using a substitution when |k| < 1. The final solution involves using trigonometric identities and an understanding of the dirac-delta function to arrive at the answer θ(1-|k|) * 2/sqrt(1-k2).
  • #1
zheng89120
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Homework Statement



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02∏ δ(sin θ - k) dθ

equals

θ(1-|k|) * 2/sqrt(1-k2)

Homework Equations



integrating dirac-delta function

The Attempt at a Solution



[sin (2pi) - k] - [sin 0 - k] ??
 
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  • #2
now I'm assuming the second theta is actually a heaviside function (some bad choice of notation but surprisingly to a few other recent posts)

as for your answer it doesn't make a heap of sense, what do you actually know about integrating the delta function...
 
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  • #3
first consider when |k| > 1,what is the value of the integral...?
 
  • #4
now when |k| < 1, i would try a substitution... and ask yourself again what do you actually know about integrating the delta function?

the form of the answer may also give u a hint, though I haven't fully worked it through... what common function f(k) integrates to 1/(1-k^2)?
 
  • #5
I am not sure how to integrate dirac-delta of a polynomial such as [sin(theta) - k] unfortunately.
 
  • #6
Do you know what [itex]\int_a^b \delta(x) dx[/itex] itself is?
 
  • #7
exactly start from what you do know
 
  • #8
zheng89120 said:
I am not sure how to integrate dirac-delta of a polynomial such as [sin(theta) - k] unfortunately.
also that is not a polynomial
 
  • #9
HallsofIvy said:
Do you know what [itex]\int_a^b \delta(x) dx[/itex] itself is?

Once you can answer that ask the following questions, how do you solve
[itex]\int_a^b \delta(2x) dx[/itex]

then
[itex]\int_a^b \delta(2x-k) dx[/itex]

and what is the value of
[itex]\int_a^b \delta(2x-k) f(x) dx[/itex]

If you can do those you shoudl have a pretty good idea how to do approach the problem, along with some trig and you should be there
 
  • #10
okay, i think i got it, or at least as long as I am assuming a trig identity correctly, much thanks
 

FAQ: What is the Integral of a Dirac-Delta Function with a Sine Argument?

What is the dirac-delta function?

The dirac-delta function, commonly denoted as δ(x), is a mathematical function that is used to represent a point mass or impulse at a specific point in a function. It is zero everywhere except at x=0, where it is infinite. It is often used in physics and engineering to represent concentrated forces or impulses.

What is the purpose of integrating a dirac-delta function?

The main purpose of integrating a dirac-delta function is to find the value of the function at the point where the delta function is located. This is often used in solving differential equations and evaluating complex integrals.

How do you integrate a dirac-delta function?

The integration of a dirac-delta function can be done using the sifting property, which states that the integral of δ(x) multiplied by any function f(x) is equal to f(0). This means that the integral of the dirac-delta function results in the value of the function at x=0. In other words, the integral of the dirac-delta function is equal to 1.

Can the dirac-delta function be integrated with other functions?

Yes, the dirac-delta function can be integrated with other functions. This is known as the convolution of the dirac-delta function with the other function. The result of this integration is a new function that describes the response of a system to an impulse input.

Are there any applications of integrating a dirac-delta function?

Yes, integrating a dirac-delta function has many practical applications in various fields such as physics, engineering, and mathematics. It is used in solving differential equations, evaluating complex integrals, and representing impulse inputs in systems. It is also used in signal processing, where it helps in analyzing and filtering signals.

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