SpecialFuncs.EllipComplete Method

Complete elliptic integral of the first kind, K(m).

Computes the complete elliptic integral of the first kind in the parameter convention m = k^2 (m is the modulus squared, not the
modulus k), evaluated by the arithmetic-geometric-mean (AGM) iteration:

K(m)=integral _0^(pi/2)dtheta/(sqrt(1-m sin^2theta))=pi/(2 AGM(1,sqrt(1-m))).

Domain: the parameter m must lie on the closed interval [0,1]. Boundary behavior: K(0)=\pi/2; at m=1 the integral diverges and the routine returns +INF. For m<0 or m>1 the argument is outside the domain and an exception is raised (it does not return NaN).

Complete elliptic integrals of the first and second kind, K(m) and E(m).

Returns the complete elliptic integral of the first kind K(m) and, through the out-parameter e, the complete elliptic integral of the second kind E(m), both in the parameter convention m = k^2, computed by a shared AGM iteration to accuracy Epsilon:

K(m)=integral _0^(pi/2)dtheta/(sqrt(1-m sin^2theta)), E(m)=integral _0^(pi/2)sqrt(1-m sin^2theta) dtheta.

Domain: m on the closed interval [0,1]. Boundary behavior: K(0)=E(0)=\pi/2; at m=1, K=+INF and E=1. For m<0 or m>1 an exception is raised (no NaN return).