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Math Help - Elliptic integral power series expansion

  1. #1
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    Elliptic integral power series expansion

    Hi,

    I've attached a problem I've come across and was wondering if anyone could give me a hand?

    Cheers.
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  2. #2
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    Re: Elliptic integral power series expansion

    You need to evaluate I_k = \int_0^{\pi/2}\sin^{2k}\phi\,d\phi. Integrate by parts:

    \begin{aligned}I_k &= \int_0^{\pi/2}\sin\phi\sin^{2k-1}\phi\,d\phi \\ &= \Bigl[-\cos\phi\sin^{2k-1}\phi\Bigr]_0^{\pi/2} + \int_0^{\pi/2}(2k-1)\cos^2\phi\sin^{2k-2}\phi\,d\phi \\ &= \int_0^{\pi/2}(2k-1)(1-\sin^2\phi)\sin^{2k-2}\phi\,d\phi = (2k-1)(I_{k-1}-I_k).\end{aligned}

    Therefore I_k = \frac{2k-1}{2k}I_{k-1} = \frac{k-\frac12}kI_{k-1}.

    Now check that I_1=\pi/4 and use induction to show that I_k = \frac{\pi}2\,\frac{\bigl(\frac12\bigr)_k}{k!}.
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