NAME¶

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

Synopsis¶

Defined in header <cmath>
float ellint_3 ( float k, float nu, float phi );

double ellint_3 ( double k, double nu, double phi ); (since C++17)
(until C++23)
long double ellint_3 ( long double k, long double nu, long
double phi );
/* floating-point-type */ ellint_3( /* floating-point-type */
k,

/* floating-point-type */ (since C++23)
nu,

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

/* common-floating-point-type */ (A) (since C++17)

ellint_3( Arithmetic1 k, Arithmetic2 nu, Arithmetic3 phi
);

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), (until C++23)
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)).
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 (since C++23)
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.

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 (complete) elliptic integral of the third kind
(C++17) (function)
(C++17)
(C++17)

External links¶

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

Category:¶

* Todo with reason