Elliptic integral problems in mma

• To: mathgroup at yoda.physics.unc.edu
• Subject: Elliptic integral problems in mma
• From: Ari Lehtonen <lehtonen at kalikka.jyu.fi>
• Date: Mon, 26 Oct 1992 15:27:37 +0200

```While preparing notes for my course on ODE, I found a bug
in mma's elliptic integrals (Mathematica 2.0 for SPARC).

JacobiAmplitude should give the solution to the eqution
of the pendulum:
y'' + g/l Sin[y] == 0
initial cond: y[0]==0, y'[0]==z0 > 0
==> y[t] == 2 JacobiAmplitude[ z0 t/2 , 4g/(z0^2 l)]

But the plotted result didn't look very nice.
You can see problems in the following:

In[2]:= m=1.5;

In[3]:= Plot[JacobiAmplitude[x,m],{x,0,7}];

In[4]:= Plot[ArcSin[JacobiSN[x,m]],{x,0,7}];

The above curves should be the same (and are not on my mma).
Similarly the following two curves differ (while they shouldn't):

In[5]:= Plot[{Sin[JacobiAmplitude[x,m]], JacobiSN[x,m]},{x,0,7}];

Note that the parameter m > 1 (periodic solution).
The case m<1 corresponds to a strictly increasing
y=y[t] (initial push z0 large enough), and in this
case there is only one curve in Out[5].

Can anybody point out what is wrong.

Ari Lehtonen
/==================================================\
| Dept. of Mathematics | Internet: lehtonen at jyu.fi |
| Univ. of Jyvaskyla   | Phone   : +358 41 602 718 |
| P.O Box 35           | Fax     : +358 41 602 701 |