#include <cmath>
#include <iostream>

int main()
{
    const auto pi = std::acos(-1.0);
    const auto k  = 0.0;
    
    std::cout << "nu\tincomp.\tcompl.\texpect\n";
    for (auto nu : {-0.9, -0.5, 0.0, 0.5, 0.9})
    {
        auto expected = pi/(2*std::sqrt(1-nu));
        std::cout 
            << nu << '\t' 
            << std::ellint_3(k, nu, pi/2) << '\t'
            << std::comp_ellint_3(k, nu) << '\t'
            << expected << '\n';                
    }
}