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Math Help - elliptic integral of SQRT(cosx)

  1. #1
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    elliptic integral of SQRT(cosx)

    hey..we are studying elliptic integrals in my class and I did some additional research online....where i came across the formula for the integral of SQRT(cosx) = 2E (1/2 *x, SQRT(2)). I couldn't figure out how to get there myself...would someone mind writing out the proof of this?
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  2. #2
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    If we define the elliptic integral of the second kind as:

    \textbf{E}(\phi,k)=\int_{0}^{\phi} \sqrt{1-k^2\sin(\theta)}d\theta

    and we can write (using half-angle formula, then sub u=\theta/2):

    \int_0^x \sqrt{\cos(\theta)}d\theta=2\int_{0}^{x/2}\sqrt{1-2\sin^2(u)} du

    then:

    2\int_{0}^{x/2}\sqrt{1-2\sin^2(u)} du=2\textbf{E}(x/2,\sqrt{2})
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  3. #3
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    is there a way to write it in the forms of both the first and second kinds of elliptic integrals?

    (F is the same as E but d(theta) is on the top of a fraction in the integral, with the rest on the bottom of the fraction
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  4. #4
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    Quote Originally Posted by minivan15 View Post
    is there a way to write it in the forms of both the first and second kinds of elliptic integrals?

    (F is the same as E but d(theta) is on the top of a fraction in the integral, with the rest on the bottom of the fraction
    I don't see how it can be written in terms of elliptical integrals of the first kind:

    \textbf{F}(\phi,k)=\int_0^{\phi}\frac{d\theta}{\sq  rt{1-k^2\sin^2(\theta)}}
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