# Triple Integral

• Jan 17th 2010, 03:43 AM
dedust
Triple Integral
Find a condition on $G$ meeting the requirement that the equation

$M = 4G \int \int \int_{\Omega} \frac{d^3p}{(2\pi)^3} \frac{M}{\sqrt{p^2 + M^2}}$

possesses a solution for $M > 0$, where the integration is carried out over the region $\Omega$, a sphere with radius $r$ in the $p$ space.

use the fact : $\int \sqrt{x^2 + a} ~dx = \frac{1}{2} \left[x\sqrt{x^2 + a} + a \ln \left(x + \sqrt{x^2 + a}\right) \right]$ for $a > 0$

I really don't have any idea to start doing this problem.
any comment would be appreciated.

:)
• Jan 17th 2010, 09:11 AM
CaptainBlack
Quote:

Originally Posted by dedust
Find a condition on $G$ meeting the requirement that the equation

$M = 4G \int \int \int_{\Omega} \frac{d^3p}{(2\pi)^3} \frac{M}{\sqrt{p^2 + M^2}}$

possesses a solution for $M > 0$, where the integration is carried out over the region $\Omega$, a sphere with radius $r$ in the $p$ space.

use the fact : $\int \sqrt{x^2 + a} ~dx = \frac{1}{2} \left[x\sqrt{x^2 + a} + a \ln \left(x + \sqrt{x^2 + a}\right) \right]$ for $a > 0$

I really don't have any idea to start doing this problem.
any comment would be appreciated.

:)

Obscure notation (at least from my experience) obviously $d^3p$ is meant to be a volume element but what is $p^2$?

CB