Michel Talon wrote:
> Raymond Toy wrote:
>
>
>>>
>>>
>> FWIW, I tried implementing the reduction algorithm in A&S for converting
>> R(x,q) to elliptic integals of the first, second, and third kind. Parts
>> of it work, but many parts do not. In particular, I never got it to
>> recognize that sqrt((1-x^2)/(1-m*x^2)) is an elliptic integral.
>>
>> Ray
>>
>
> Hi,
>
> thank you for your comments, Raymond and Richard. The example above is the
> occasion to show how to reduce such things to canonical form.
>
> First one writes sqrt((1-x^2)/(1-m*x^2)) = 1/(1-mx^2) sqrt((1-x^2)(1-mx^2))
>
Hmm. I think I would have converted that to
(1-x^2)/sqrt((1-x^2)*(1-m*x^2)), but I guess either way works. I just
never got around to teaching maxima to recognize and convert that form.
My algorithms generally converted cubic forms to quartic forms, since
that seemed to be how A&S was handling the integrals.
Oh, and my stuff was not integrated into the integration routines. You
had to call the routine manually with a known elliptic integral form.
>
> By the way, i think one of the problems when trying to compute elliptic
> integrals is trying to coerce everything to old Jacobian elliptic
> notations, cn, sn etc. which are in fact extremely unnatural from the point
>
At least for maxima, Jacobian elliptic functions are more useful because
maxima actually knows something about them. Maxima doesn't know
anything about the Weierstrass p function. :-)
Ray