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Math Help - please if you know about elliptic integrals I NEED help!

  1. #1
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    please if you know about elliptic integrals I NEED help!

    [integral from 0 to pi/2] dx/SQRT(a - cosx)

    (a >1)

    I need to write this in terms of the elliptic integrals F

    the elliptic integral of the first kind is F(k,q) = [integral from 0 to q]
     <br />
du/SQRT(1 - (ksinu)^2)<br />
    PLEASE if you know anything about elliptic integrals I need your help!

    I do not see a starting point on this!

    I know elliptic integrals are tricky and ambiguous and all that...I merely need to REWRITE the given integral in terms of F

    thank you in advance for your help!
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  2. #2
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    If \int_{0}^{\pi/2}\frac{dx}{\sqrt{a-\cos(x)}}=\frac{2}{\sqrt{a-1}}\int_0^{\pi/4}\frac{du}{\sqrt{1-\frac{2}{a-1}\sin^2(u)}}

    can you use the identity \sin^2(x/2)=\frac{1-\cos(x)}{2} to get there? Just make the substitution for cos(x), and another one u=x/2.
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  3. #3
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    using your idea:

    SQRT(a -(1 - 2(sinx/2)^2) = SQRT( a - 1 + 2(sinx/2)^2)

    then factoring out a - 1 gives SQRT(a-1) * SQRT(1 + 2/(a-1)*(sinx/2)^2)

    where k^2 is both negative and not necessarily less than one...the conditions for k in elliptic integrals are 0< k <1

    did I do something wrong, or is the idea slightly off? I tried playing around with the concept but didn't get anywhere. Is there another answer or is this one correct but I'm just doing it incorrectly?
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  4. #4
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    I'm not sure. Sorry. Looks then to me a needs to be greater than 3.
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  5. #5
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    Re: please if you know about elliptic integrals I NEED help!

    I want to solve this eqn using the elliptic integral
    int(1/sqrt(1-ksin(phi)))
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  6. #6
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    Re: please if you know about elliptic integrals I NEED help!

    Quote Originally Posted by prash View Post
    I want to solve this eqn using the elliptic integral
    int(1/sqrt(1-ksin(phi)))
    do you mean

    $$\int \frac{1}{\sqrt{1- k\sin(\phi)}}d\phi$$

    if so it's

    $$-\frac{2 F\left(\frac{1}{4}(\pi-2\phi),\frac{2k}{k-1}\right) \sqrt{\frac{k\sin(\phi)-1}{k-1}}}{\sqrt{1-k\sin(\phi)}}$$

    $$F(\phi,m)\mbox{ is the elliptic integral of the first kind.}$$
    Last edited by romsek; February 18th 2014 at 02:07 AM.
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  7. #7
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    Re: please if you know about elliptic integrals I NEED help!

    I don understand about the reversal of polarity statement.I am new to elliptic integrals,i do have one doubt.whether i can get a variable solution when i use elliptical integral or only numerical solution is possible?
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  8. #8
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    Re: please if you know about elliptic integrals I NEED help!

    my mistake, getting the tex correct was a pain, it's corrected.
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  9. #9
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    Re: please if you know about elliptic integrals I NEED help!

    @Romsek,Can u elaborate me with the proof for the solution
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  10. #10
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    Re: please if you know about elliptic integrals I NEED help!

    nope, I just plugged it into Mathematica and that's what came out. sorry
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  11. #11
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    Re: please if you know about elliptic integrals I NEED help!

    how can i have a numerical solution for the above solution?
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  12. #12
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    Re: please if you know about elliptic integrals I NEED help!

    If you have a new question, you should start a new thread. From the main calculus page, it's under Forum Tools -> Post a new Thread.

    - Hollywood
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