#include <cmath>

  double comp_ellint_3(double k, double nu)  // C++17

Liefert Wert des vollständigen Elliptischen Integrals 3. Art $\Pi(k,\nu) = \int_0^{\pi/2} \frac{d\theta}{(1-\nu \sin^2 \theta)\sqrt{1-k^2 \sin^2 \theta}} = \Pi'(k,\nu,\pi/2)$.

Rückgabewert: $\Pi(k,\nu)$.

ellint_1(), ellint_2(), ellint_3(), comp_ellint_1(), comp_ellint_2().

ellint_3.cpp

#include <cmath>
#include <iostream>
 
int main()
{
  std::cout << "# k n=-3 n=-0.5 n=0 n=0.5\n";
 
  for (int i = 0; i < 100; ++i)
  {
    double k = 0.01*i;
    std::cout << k 
      << '\t' << std::comp_ellint_3(k, -3.0) 
//       << '\t' << std::comp_ellint_3(k, -1.0)  // inf
      << '\t' << std::comp_ellint_3(k, -0.5) 
      << '\t' << std::comp_ellint_3(k, 0.0) 
      << '\t' << std::comp_ellint_3(k, 0.5)
//      << '\t' << std::comp_ellint_3(k, 1.0) // domain error
//      << '\t' << std::comp_ellint_3(k, 3.0) // domain error
      << '\n';
  }
}

Anmerkung: Die gcc-Bibliothek verwendete bis gcc 7.2.0 das entgegengesetzte Vorzeichen für die Charakteristik $\nu$ (GCC bug report 66689). Ab gcc 8.0.0 HEAD 201711 scheint dies behoben (C++2a, Wandbox):

ellint3_test.cpp

#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';                
    }
}