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]

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_{0}^{ϕ} \frac{d θ}{(1 - ν {sin}^{2} θ) \sqrt{1 - k^{2} {sin}^{2} θ}}, for | k | \leq 1 \\ (29.34) \end{matrix}