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Math Help - Triple Integral

  1. #1
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    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.

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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by dedust View Post
    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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