[sf.cmath.ellint.3]

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floating-point-type ellint_3(floating-point-type k, floating-point-type nu,
                             floating-point-type phi);
float        ellint_3f(float k, float nu, float phi);
long double  ellint_3l(long double k, long double nu, long double phi);

Effects: These functions compute the incomplete elliptic integral of the third kind of their respective arguments k, nu, and phi (phi measured in radians).

Returns:

Π (ν, k, ϕ), where Π is given by Formula 29.34, ν is nu, k is k, and φ is phi .

\begin{matrix} Π (ν, k, ϕ) = {\int ϕ 0 d θ (1 - ν {sin 2 θ) \sqrt{1 - {k 2 {sin 2 θ}^{2 θ}}^{2 {sin 2 θ}^{2 θ}}}}^{2 θ) \sqrt{1 - {k 2 {sin 2 θ}^{2 θ}}^{2 {sin 2 θ}^{2 θ}}}}}_{0 d θ (1 - ν {sin 2 θ) \sqrt{1 - {k 2 {sin 2 θ}^{2 θ}}^{2 {sin 2 θ}^{2 θ}}}}^{2 θ) \sqrt{1 - {k 2 {sin 2 θ}^{2 θ}}^{2 {sin 2 θ}^{2 θ}}}}}^{ϕ 0 d θ (1 - ν {sin 2 θ) \sqrt{1 - {k 2 {sin 2 θ}^{2 θ}}^{2 {sin 2 θ}^{2 θ}}}}^{2 θ) \sqrt{1 - {k 2 {sin 2 θ}^{2 θ}}^{2 {sin 2 θ}^{2 θ}}}}} \end{matrix}

29 Numerics library [numerics]

29.7 Mathematical functions for floating-point types [c.math]

29.7.6 Mathematical special functions [sf.cmath]

29.7.6.14 Incomplete elliptic integral of the third kind [sf.cmath.ellint.3]