Elliptic functions

**juanfe_zodiac** · October 20th 2008, 08:08 PM

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.

**shawsend** · October 21st 2008, 08:09 AM

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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