--Barton
maxima-bounces at math.utexas.edu wrote on 04/14/2011 11:25:44 AM:
> (%i3) hyper_int(1/sqrt(x*(x+1)*(x-1)),x);
> ;
>
> ; Warning: This function is undefined:
> ; GAMMA
> 1 1 5 2 2
> 2 hypergeometric([-, -], [-], x ) x sqrt(1 - x )
> 4 2 4
> (%o3) ------------------------------------------------
> 3
> sqrt(x - x)
>
> First the gamma function is well defined in maxima, and second the
> integral in question is obviously elliptic, and i have not found formula
> which reduces incomplete elliptic integral to hypergeometric functions
> (in contrast with complete integral). Maybe there exists one, but i am
not
> aware.
There is also inverse_jacobi_int:
(%i124) inverse_jacobi_int(1/sqrt(x*(x+1)*(x-1)),x);
(%o124)
-(2*%i*inverse_jacobi_sn((%i*sqrt(x-1))/sqrt(2),2)*sqrt(x-1)*sqrt(x)*sqrt(x+1))/sqrt(x^3-x)
Check:
(%i125) ratsimp(diff(%,x));
(%o125) 1/sqrt(x^3-x)
(%i126) hyper_int(1/sqrt(x*(x+1)*(x-1)),x);
(%o126)
(2*hypergeometric([1/4,1/2],[5/4],x^2)*x*sqrt(1-x^2))/sqrt(x^3-x)
This might also be correct, but I don't know which recursions to use to
show it.
Evidence that %o126 might be correct:
(%i127) taylor(diff(%,x) - 1/sqrt(x*(x+1)*(x-1)),x,0,128);
(%o127)/T/ 0+...
I don't get the warning about gamma--maybe this is due to the Maxima
version you
are using?
--bw