# std::comp_ellint_3, std::comp_ellint_3f, std::comp_ellint_3l

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Defined in header `<cmath>` |  |
| (1) |
float comp_ellint_3 ( float k, float nu ); double comp_ellint_3 ( double k, double nu ); long double comp_ellint_3 ( long double k, long double nu ); |  | (since C++17) (until C++23)
/* floating-point-type */ comp_ellint_3( /* floating-point-type */ k, /* floating-point-type */ nu ); |  | (since C++23)
float comp_ellint_3f( float k, float nu ); | (2) | (since C++17)
long double comp_ellint_3l( long double k, long double nu ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Arithmetic1, class Arithmetic2 > /* common-floating-point-type */ comp_ellint_3( Arithmetic1 k, Arithmetic2 nu ); | (A) | (since C++17)

1-3) Computes the complete elliptic integral of the third kind of the arguments k and nu. The library provides overloads of `std::comp_ellint_3` for all cv-unqualified floating-point types as the type of the parameters k and nu. (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)

### Return value

If no errors occur, value of the complete elliptic integral of the third kind of k and nu, that is std::ellint_3(k, nu, π/2), 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 and second argument num2:

If num1 or num2 has type long double, then std::comp_ellint_3(num1, num2) has the same effect as std::comp_ellint_3(static_cast<long double>(num1), static_cast<long double>(num2)). Otherwise, if num1 and/or num2 has type double or an integer type, then std::comp_ellint_3(num1, num2) has the same effect as std::comp_ellint_3(static_cast<double>(num1), static_cast<double>(num2)). Otherwise, if num1 or num2 has type float, then std::comp_ellint_3(num1, num2) has the same effect as std::comp_ellint_3(static_cast<float>(num1), static_cast<float>(num2)). | (until C++23)
If num1 and num2 have arithmetic types, then std::comp_ellint_3(num1, num2) has the same effect as std::comp_ellint_3(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2)), where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2, 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>

    int main()
    {
        std::cout << std::fixed
                  << "Π(0.5,0) = " << std::comp_ellint_3(0.5, 0) << '\n'
                  << "K(0.5)   = " << std::comp_ellint_1(0.5) << '\n'
                  << "Π(0,0)   = " << std::comp_ellint_3(0, 0) << '\n'
                  << "π/2      = " << std::acos(-1) / 2 << '\n'
                  << "Π(0.5,1) = " << std::comp_ellint_3(0.5, 1) << '\n';
    }

Output:

    Π(0.5,0) = 1.685750
    K(0.5)   = 1.685750
    Π(0,0)   = 1.570796
    π/2      = 1.570796
    Π(0.5,1) = inf

### See also

ellint_3 ellint_3f ellint_3l (C++17)(C++17)(C++17) | (incomplete) 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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