We use EllipticPi when we write exact solutions of rotation of a free asymmetric top. While solving Euler’s equations for angular velocity or angular momentum in the body frame we need Jacobi elliptic functions solving the differential equation for the attitude matrix involves EllipticPi function. As I have explained it in Taming the T-handle continued we need the integral
In Mathematica this is easily implemented as
While the documentations of both Mathematica and Maple contain links to Abramowitz and Stegun Handbook of Mathematical Functions, they use different definitions. Here is what concerns us, from p. 590 of the 10th printing:
What we need is 17.2.16, while Maple is using 17.2.14. To convert we need to set but such a conversion is possible only in the domain where can be inverted. We can do it easily for sufficiently small values of but not necessarily for values that contain several quarter-periods.
The following Maple procedure does the job:
HAs an example here is the Maple plot for nu=-3, k=0.9:
plot(('epi')(t, -3.0, .9), t = -20 .. 20)
And here is the corresponding Mathematica plot:
The function epi(t,nu, k) defined above for Maple gives now the same result as EllipticPi(nu,JacobiAM(t,k^2),k^2) in Mathematica.