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

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Mathematical 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)
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 ∫ phi0
dθ
(1-nusin 2 θ) √ 1-k 2 sin 2 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

(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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