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Thread: Elliptic functions

  1. #1
    Junior Member
    Oct 2008

    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
    Aug 2008
    Elliptic functions are inverses of the elliptic integrals. Here's an elliptic integral:

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

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

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

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

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

    When you solve this equation, you'll get an elliptic integral in terms of $\displaystyle \theta$ which of course you'll want to solve for $\displaystyle \theta$ 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:

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

    Where $\displaystyle k$ is a constant created in the calculations.
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