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

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< cpp‎ | numeric‎ | special functions

Defined in header `<cmath>` |  |
| (1) |
float ellint_3 ( float k, float nu, float phi ); double ellint_3 ( double k, double nu, double phi ); long double ellint_3 ( long double k, long double nu, long double phi ); |  | (since C++17) (until C++23)
/* floating-point-type */ ellint_3( /* floating-point-type */ k, /* floating-point-type */ nu, /* floating-point-type */ phi ); |  | (since C++23)
float ellint_3f( float k, float nu, float phi ); | (2) | (since C++17)
long double ellint_3l( long double k, long double nu, long double phi ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Arithmetic1, class Arithmetic2, class Arithmetic3 > /* common-floating-point-type */ ellint_3( Arithmetic1 k, Arithmetic2 nu, Arithmetic3 phi ); | (A) | (since C++17)

1-3) Computes the incomplete elliptic integral of the third kind of k, nu, and phi. The library provides overloads of `std::ellint_3` for all cv-unqualified floating-point types as the type of the parameters k, nu and phi. (since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

### Parameters

k | - | elliptic modulus or eccentricity (a floating-point or integer value)
nu | - | elliptic characteristic (a floating-point or integer value)
phi | - | Jacobi amplitude (a floating-point or integer value, measured in radians)

### Return value

If no errors occur, value of the incomplete elliptic integral of the third kind of k, nu, and phi, that is ∫phi
0

dθ
(1-nusin2 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.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their first argument num1, second argument num2 and third argument num3:

If num1, num2 or num3 has type long double, then std::ellint_3(num1, num2, num3) has the same effect as std::ellint_3(static_cast<long double>(num1), static_cast<long double>(num2), static_cast<long double>(num3)). Otherwise, if num1, num2 and/or num3 has type double or an integer type, then std::ellint_3(num1, num2, num3) has the same effect as std::ellint_3(static_cast<double>(num1), static_cast<double>(num2), static_cast<double>(num3)). Otherwise, if num1, num2 or num3 has type float, then std::ellint_3(num1, num2, num3) has the same effect as std::ellint_3(static_cast<float>(num1), static_cast<float>(num2), static_cast<float>(num3)). | (until C++23)
If num1, num2 and num3 have arithmetic types, then std::ellint_3(num1, num2, num3) has the same effect as std::ellint_3(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2), static_cast</* common-floating-point-type */>(num3)), where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank among the types of num1, num2 and num3, arguments of integer type are considered to have the same floating-point conversion rank as double. If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided. | (since C++23)

### Example

Run this code

    #include <cmath>
    #include <iostream>
    #include <numbers>

    int main()
    {
        const double hpi = std::numbers::pi / 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?

### 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)

### External links

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

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