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Math Help - Elliptic functions

  1. #1
    Junior Member
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    Elliptic functions

    hello everyone...i need to have a mayor idea of what are the elliptic functions???...(i dont know what are they at all)some graphs will be helpfull...where does this function come from???...what kind of math is that???....thats why i need i mayor idea (to be able to create a diagram and make connection with the commun function of high school like polynomials, inverse, racionals)...tksss.
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  2. #2
    Super Member
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    Elliptic functions are inverses of the elliptic integrals. Here's an elliptic integral:

    u=F(\phi,k)=\int_0^{\phi}\frac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}

    Alright, let's take the inverse of that function, that is, for some value of u with fixed k, figure out what \phi has to be. Well, that's easy, it's the elliptic function ( \textbf{su}):

    F^{-1}(\phi,k)=\textbf{su}(u,k)

    One application of elliptic functions is in the solution of the non-linear equation for a pendulum:

    \frac{d^2\theta}{dt^2}+\frac{g}{L}\sin(\theta)=0

    When you solve this equation, you'll get an elliptic integral in terms of \theta which of course you'll want to solve for \theta right? Well, you'll have to ``invert'' the expression by taking the inverse of that integral and that results in the elliptic function. The solution is then:

    \theta=2\arcsin\left[k\; \textbf{sn}\left(t\sqrt{g/L},k\right)\right].

    Where k is a constant created in the calculations.
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