#include <cmath>

  double ellint_3(double k, double nu, double phi)  // C++17

Liefert Wert des unvollständigen Elliptischen Integrals 3. Art $\Pi(k,\nu,\varphi) = \int_0^\varphi \frac{d\theta}{(1-\nu \sin^2 \theta)\sqrt{1-k^2 \sin^2 \theta}}$ für $|k| \leq 1$ und $\nu > -1/\sin^2 \theta$.

Rückgabewert: $\Pi(k,\nu,\varphi)$.

ellint_1(), ellint_2(), comp_ellint_1(), comp_ellint_2(), comp_ellint_3().

ellint_3.cpp

#include <cmath>
#include <iostream>
 
int main()
{
  const auto pi = std::acos(-1.0);
 
  std::cout << "# k=0,n=0 k=0,n=0.5 k=1,n=0.5\n";
 
  for (int i = 0; i < 100; ++i)
  {
    double phi = 0.01*i*pi/2;
    std::cout << phi 
      << '\t' << std::ellint_3(0.0, 0.0, phi)
      << '\t' << std::ellint_3(0, 0.5, phi)
      << '\t' << std::ellint_3(1, 0.5, phi)
      << '\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';                
    }
}