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

 

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Mathematical special functions

assoc_laguerreassoc_laguerrefassoc_laguerrel   assoc_legendreassoc_legendrefassoc_legendrel betabetafbetal comp_ellint_1comp_ellint_1fcomp_ellint_1l comp_ellint_2comp_ellint_2fcomp_ellint_2l comp_ellint_3comp_ellint_3fcomp_ellint_3l cyl_bessel_icyl_bessel_ifcyl_bessel_il

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl   cyl_bessel_kcyl_bessel_kfcyl_bessel_kl cyl_neumanncyl_neumannfcyl_neumannl ellint_1ellint_1fellint_1l ellint_2ellint_2fellint_2l ellint_3ellint_3fellint_3l expintexpintfexpintl

hermitehermitefhermitel laguerrelaguerreflaguerrel legendrelegendreflegendrel riemann_zetariemann_zetafriemann_zetal sph_besselsph_besselfsph_bessell sph_legendresph_legendrefsph_legendrel sph_neumannsph_neumannfsph_neumannl

 

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

See also

comp_ellint_3comp_ellint_3fcomp_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.