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

[zheng89120]

Homework Statement

Show:

₀∫^2∏ 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] ??

 

[lanedance]

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...

 

[lanedance]

first consider when |k| > 1,what is the value of the integral...?

 

[lanedance]

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)?

 

[zheng89120]

I am not sure how to integrate dirac-delta of a polynomial such as [sin(theta) - k] unfortunately.

 

[HallsofIvy]

Do you know what \int_a^b \delta(x) dx itself is?

 

[lanedance]

exactly start from what you do know

 

[lanedance]

I am not sure how to integrate dirac-delta of a polynomial such as [sin(theta) - k] unfortunately.

also that is not a polynomial

 

[lanedance]

Do you know what \int_a^b \delta(x) dx itself is?

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

then
\int_a^b \delta(2x-k) dx

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

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

 

[zheng89120]

okay, i think i got it, or at least as long as I am assuming a trig identity correctly, much thanks