# std::ellint_3, std::ellint_3f, std::ellint_3l

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< cpp‎ | numeric‎ | special math

double ellint_3( double k, double ν, double φ ); float ellint_3f( float k, float ν, float φ ); long double ellint_3l( long double k, long double ν, long double φ ); | (1) | (since C++17)
Promoted ellint_3( Arithmetic k, Arithmetic ν, Arithmetic φ ); | (2) | (since C++17)

1) Computes the incomplete elliptic integral of the third kind of `k`, `ν`, and `φ`.
2) A set of overloads or a function template for all combinations of arguments of arithmetic type not covered by (1). If any argument has integral type, it is cast to double. If any argument is long double, then the return type `Promoted` is also long double, otherwise the return type is always double.

### Parameters

k | - | elliptic modulus or eccentricity (a value of a floating-point or integral type)
ν | - | elliptic characteristic (a value of floating-point or integral type)
φ | - | Jacobi amplitude (a value of floating-point or integral type, measured in radians)

### Return value

If no errors occur, value of the incomplete elliptic integral of the third kind of `k`, `ν`, and `φ`, that is ∫φ
0

dθ
(1-νsin2 θ)√1-k2 sin2 θ

, is returned.

### Error handling

Errors may be reported as specified in math_errhandling

- If the argument is NaN, NaN is returned and domain error is not reported
- If |k|>1, a domain error may occur

### Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header `tr1/cmath` and namespace `std::tr1`

An implementation of this function is also available in boost.math

### Example

Run this code

    #include <cmath>
    #include <iostream>
    int main()
    {
        double hpi = std::acos(-1)/2;
        std::cout << "Π(0,0,π/2) = " << std::ellint_3(0, 0, hpi) << '\n'
                  << "π/2 = " << hpi << '\n';
    }

Output:

    Π(0,0,π/2) = 1.5708
    π/2 = 1.5708

| This section is incomplete Reason: this and other elliptic integrals deserve better examples.. perhaps calculate elliptic arc length?

### External links

Weisstein, Eric W. "Elliptic Integral of the Third Kind." From MathWorld--A Wolfram Web Resource.

### See also

comp_ellint_3 comp_ellint_3f comp_ellint_3l (C++17)(C++17)(C++17) | (complete) elliptic integral of the third kind (function)

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- Todo with reason