Complicated Definite Double Integral

  1. I've been working on a problem involving a large meteoroid passing over the Earth and what its gravitational effects would be on the Earth's mantle. I developed an equation for this, and I've worked it down to a certain point, but unfortunately, I'm not sure how to finally solve it. By the way, I downloaded a free trial version of Mathematica, but it doesn't seem to be able to solve the problem, either.

    The equation is as follows:

    Math Eq.jpg

    (I'm new here, so I didn't know the best way to input an equation.)
    Any help that could be provided would be very much appreciated.

    - Brandon
  2. jcsd
  3. LeonhardEuler

    LeonhardEuler 863
    Gold Member

    The "r" part of the integral is doable. I get
    [tex]P=\int_{0}^{\pi}2\rho Gmdr^{2}a\sin{a}\dot ( \ln{\sqrt{r^{2}+d^{2}-\cos{a}} +r} -\frac{r}{\sqrt{r^{2}+d^{2}-\cos{a}}})da\left|^{r=6.275*10^6}_{r=6.175*10^6}[/tex]

    Integrating the "a" part seems like it should be a nightmare, though. You could do a substitution u=-cos(a) if it wasn't for that a sitting outside. It's probably not expressible in elementary functions. Could you just evaluate it numerically?
  4. AlephZero

    AlephZero 7,244
    Science Advisor
    Homework Helper

    Starting with the a integral,

    [tex]\int \frac{a \sin a\, da}{(b^2 - \cos a)^{3/2}}[/tex]

    is an elliptic function of the 3rd kind according to

    My knowledge of elliptic functions doesn't extend to knowing if your definite integral equals something nice, but at least that's a start.
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