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## elliptic integrals

The EllipticK(k) function returns the complete elliptic integral of the first kind, i.e. the definite integral between 0 and pi/2 of the function (1 - k^2*sin^2(θ))^(-0.5). The domain of k is -1 to 1 (exclusive).

The EllipticE(k) function returns the complete elliptic integral of the second kind, i.e. the definite integral between 0 and pi/2 of the function (1 - k^2*sin^2(θ))^0.5. The domain of k is -1 to 1 (inclusive).

The EllipticPi(n,k) function returns the complete elliptic integral of the third kind, i.e. the definite integral between 0 and pi/2 of the function (1 - k^2*sin^2(θ))^(-0.5) / (1 - n*sin^2(θ)). The parameter n must be less than 1, while k must lie between -1 and 1 (exclusive). Note that by definition EllipticPi(0,k) == EllipticK(k) for all possible values of k.

Elliptic integral algorithm: B.C.Carlson 1995, Numerical Algorithms 10:13-26.