Solve Troublesome Integral Homework Problem

[Appledave]

Homework Statement

Here is the problem: http://img810.imageshack.us/i/skjghl.jpg/
(I failed repeatedly to write the problem with forum latex code)

Homework Equations

N/A

The Attempt at a Solution

I thought that since the integrand is nonzero only when b - E - p = 0, the integral would be equal to all the constant terms times the area of a sphere with radius given by b - E - p = 0 (that is, p = (b^2 - a^2)/2b), but I wound up with the wrong answer.

 

[Stephen Tashi]

The use of LaTex on the forum has a bug. To get a page displayed correctly, you must use the "preview" function of the editor and then use the refresh or reload feature of your browser. Before the reload, a preview can look crazy since it picks up symbols from old messages. You must use a similar procedure when you edit a message.

$$\omega = 2 \pi G^2 \int \frac{ d^3p}{(2\pi)^3} \frac{f^2 a^2}{2E}\delta(b - E - p)$$

$$E = \sqrt{a^2 + p^2)$$ and $a,b,f$ and $G$ are constants.

Show that $$\omega = \frac{ G^2 f^2 a^2 b (1 - \frac{a^2}{b^2} )^2}{8\pi}$$

We are integrating with respect to which variables? Is $p$ a function of a single variable?

 

[Appledave]

We are integrating over all space for the single variable p.

You've made a small typo, the Dirac's delta function isn't supposed to be squared, but thanks for the heads up on the bug and for putting my question into forum latex code =].

 

[Stephen Tashi]

I edited it to fix the typo.
And I fixed another typo. I had p instead of b listed as a constant.

 

[Stephen Tashi]

I still don't understand the notation completely. This is 3D space, right? What is d ?
Is $d^3$ a constant or a notation for a derivative?

 

[Appledave]

$$d^{3}p$$ is like the $$dx$$ usually found at the end of integrals, and it means that we integrate the variable p over all of the 3D space. It should be at the end, but for some reason many people (at least among physicists) like to put it right after the integration symbol.

 

[Stephen Tashi]

How does the constraint $p = \frac{ b^2 - a^2} {2b}$ define a surface of any kind?

The letter [itex]p [/tex] denotes some function $$p(x,y,z)$$ , right? Is $$p(x,y,z) = \sqrt(x^2 + y^2 + z^2)$$[/itex]$$?$$