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Annoying trig integral

  1. Feb 22, 2010 #1
    Hi...would appreciate any suggestions re the following integral which has appeared in a celestial-mechanics calculation:

    [tex]I = \int_0^{2\pi } {\frac{1}{{(1 + e\cos \theta )^3}}d\theta } [/tex]

    where [tex]0 < e < 1[/tex].

    Integration by parts seems a sensible approach but for some reason I can't get sensible results. I presume I'm making some idiotic mistake that I'm just not picking up when I check my calculations (frustrating as hell!). I'm pretty sure there is supposed to be a fairly neat result but MATLAB and Mathematica aren't giving me anything.

    Any ideas? Thanks in advance.
  2. jcsd
  3. Feb 22, 2010 #2
    Apply the conformal transform z = exp(i theta). Then the integral becomes the integral over the unit circle in the complex plane of:

    dz/(i z) 1/[1+e (z+z^(-1))/2]^3

    Simplify the fraction and consider the poles that are inside the unit circle.
  4. Feb 23, 2010 #3
    Thank you!
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