[sf.cmath.ellint.2]

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floating-point-type ellint_2(floating-point-type k, floating-point-type phi);
float        ellint_2f(float k, float phi);
long double  ellint_2l(long double k, long double phi);

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

Returns: E(k, φ), where E is given by Formula 29.33, k is k, and φ is phi.

\begin{matrix} E (k, ϕ) = {\int ϕ 0 \sqrt{1 - {k 2 {sin 2 θ d θ, for | k | \leq 1 (29.33)}^{2 θ d θ, for | k | \leq 1 (29.33)}}^{2 {sin 2 θ d θ, for | k | \leq 1 (29.33)}^{2 θ d θ, for | k | \leq 1 (29.33)}}}}_{0 \sqrt{1 - {k 2 {sin 2 θ d θ, for | k | \leq 1 (29.33)}^{2 θ d θ, for | k | \leq 1 (29.33)}}^{2 {sin 2 θ d θ, for | k | \leq 1 (29.33)}^{2 θ d θ, for | k | \leq 1 (29.33)}}}}^{ϕ 0 \sqrt{1 - {k 2 {sin 2 θ d θ, for | k | \leq 1 (29.33)}^{2 θ d θ, for | k | \leq 1 (29.33)}}^{2 {sin 2 θ d θ, for | k | \leq 1 (29.33)}^{2 θ d θ, for | k | \leq 1 (29.33)}}}} \end{matrix}

29 Numerics library [numerics]

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

29.7.6 Mathematical special functions [sf.cmath]

29.7.6.13 Incomplete elliptic integral of the second kind [sf.cmath.ellint.2]