Triple Integral

**dedust** · January 17th 2010, 02:43 AM

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.

**CaptainBlack** · January 17th 2010, 08:11 AM

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

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