[sf.cmath.ellint.3]

# 29 Numerics library [[numerics]](./#numerics)

## 29.7 Mathematical functions for floating-point types [[c.math]](c.math#sf.cmath.ellint.3)

### 29.7.6 Mathematical special functions [[sf.cmath]](sf.cmath#ellint.3)

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

[ð](#lib: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);
`

[1](#1)

[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/numerics.tex#L10491)

*Effects*: These functions compute
the incomplete elliptic integral of the third kind
of their respective argumentsk, nu, and phi (phi measured in radians)[.](#1.sentence-1)

[2](#2)

[#](http://github.com/Eelis/draft/tree/9adde4bc1c62ec234483e63ea3b70a59724c745a/source/numerics.tex#L10498)

*Returns*: Î (Î½,k,Ï),
where Î  is given by Formula [29.34](#eq:sf.cmath.ellint.3),Î½ is nu,k is k, andÏ is phi[.](#2.sentence-1)

Î (Î½,k,Ï)=â«Ï0dÎ¸(1âÎ½sin2Î¸)â1âk2sin2Î¸Â , forÂ |k|â¤1(29.34)
