1.13.1.68 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*sin(p))**2)**(-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*sin(p))**2)**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*sin(p))**2)**(-0.5)/(1-n*sin(p)**2)‘. 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‘.