Ellipsoid surface area

[Albert Chan]

Example: Elliptic Integral of the First Kind (note: we use elliptic parameter m=k^2)

>>> from mpmath import *
>>> ellipf(2,3)
1.0010773804561062-1.490278044744527j

RFD(x,y,z) can do both (F, E) at the same time.

Code:

FE(t,m) := // return [F(t,m), E(t,m)-F(t,m)]
BEGIN
t := sin(t);
m *= t*t;
return RFD(1-t*t,1-m,1) .* [t, -m*t/3];
END;

Formula is good for angle = 0 .. pi/2

CAS> k := FE(pi/2,3)[1]      → 1.00107738046-1.17142008415*i

// 2 = 2*pi/2 + (2-pi)
CAS> 2*k + FE(2-pi,3)[1]                             → 1.00107738046-1.49027804474*i

// 2 = pi/2 + (2-pi/2)
CAS> k + FE(2-pi/2,3/(3-1))[1]/sqrt(1-3)      → 1.00107738046-1.49027804474*i