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Math Help - Help with an integral

  1. #1
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    Help with an integral

    Hi I'm doing a question regarding averaging a radial perterbation about an orbit.

    I've reduced the problem to needing to show that
    \int_0^{2\pi} \frac{\sin f}{(1+e\cos f)^{\gamma +2}} df = 0

    Where we have that 0<e<1 could I have a hint from anyone as to how to proceed with this?
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  2. #2
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    Re: Help with an integral

    Quote Originally Posted by thelostchild View Post
    Hi I'm doing a question regarding averaging a radial perterbation about an orbit.

    I've reduced the problem to needing to show that
    \int_0^{2\pi} \frac{\sin f}{(1+e\cos f)^{\gamma +2}} df = 0

    Where we have that 0<e<1 could I have a hint from anyone as to how to proceed with this?
    u = 1+e\cos{f}

    du = -e\sin{f} \, df

    substitute and reset limits of integration ...

    -\frac{1}{e}\int_{1+e}^{1+e} \frac{du}{u^{\gamma + 2}}

    note that for any function f , \int_a^a f(x) \, dx = 0
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  3. #3
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    Re: Help with an integral

    I'm afraid that method doesn't work, as the integral is non convergent for -1 < Re(e) < 1 or Im(e) != 0 so there is obviously something fishy with that substitution

    I have now figured it out, to do it (semi) rigorously, you taylor expand the (1+e*cos(f))^(-gamma-2) term which is a convergent series for -1<e<1 and integrate term by term, all of which evaluate to zero.

    Thanks for the help though
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