Richard Fateman wrote:
> Macsyma does do some elliptic integrals.
>
> Just not the ones you said..
> Here's stuff from their demo..
Thanks for these examples.
>
> (c3) INTEGRATE(1/(SQRT(1-T^2)*SQRT(2-T^2)),T,0,1);
>
> 1
> elliptic_kc(-)
> 2
> (d3) --------------
> sqrt(2)
FWIW, my current code, which is really hackish right now, isn't smart
enough to recognize this as an elliptic integral, but you can do:
(%i373) gen_ellint(1/y,t,y,1-t^2,2-t^2,t)
(%o373) inverse_jacobi_sn(t,1/2)/sqrt(2)
(%i374) limit(%,t,1)-limit(%,t,0)
(%o374) elliptic_kc(1/2)/sqrt(2)
Basically you have to write the integrand as R(x,y) where y^2 is a
quartic or cubic. Then you also have to factor the y^2 into 2 factors.
(Maxima can do the factoring, but I didn't implement it.)
> (c10) INTEGRATE(SQRT(9-T^2)/SQRT(4-T^2),T,1,2);
>
> 4 %pi 4
> (d10) 3 elliptic_ec(-) - 3 elliptic_e(---, -)
> 9 6 9
(%i390) gen_ellint((9-t^2)/y,t,y,9-t^2,4-t^2);
(%o390) 3*?%inverse_jacobi_sn(t/2,4/9)-3
*(?%inverse_jacobi_sn(t/2,4/9)
-elliptic_e(asin(t/2),4/9))
(%i391) expand(limit(%,t,2)-limit(%,t,1));
(%o391) 3*?%elliptic_ec(4/9)-3*elliptic_e(%pi/6,4/9)
(Don't know why maxima prints ?%inverse_jacobi_sn with 1D display. It's
ok with 2D display.)
If I can figure out how to get maxima to recognize elliptic integrals,
I'll try to integrate it into integrate.
Anyway, this was rather fun.
Can Macsyma do elliptic integrals of the third kind?
Ray