std::comp_ellint_1, std::comp_ellint_1f, std::comp_ellint_1l

 

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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>
double      comp_ellint_1 ( double k ); float       comp_ellint_1 ( float k ); long double comp_ellint_1 ( long double k ); float       comp_ellint_1f( float k ); long double comp_ellint_1l( long double k );	(1)	(since C++17)
double      comp_ellint_1 ( IntegralType k );	(2)	(since C++17)

1) Computes the complete elliptic integral of the first kind of k.

2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

Parameters

k

-

elliptic modulus or eccentricity (a value of a floating-point or integral type)

Return value

If no errors occur, value of the complete elliptic integral of the first kind of k, that is std::ellint_1(k, 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.

Example

The period of a pendulum of length l, given acceleration due to gravity g, and initial angle θ equals 4⋅√l/g⋅K(sin(θ/2)), where K is std::comp_ellint_1.

Run this code

#include <cmath>
#include <iostream>
#include <numbers>
 
int main()
{
    constexpr double π {std::numbers::pi};
 
    std::cout << "K(0) ≈ " << std::comp_ellint_1(0) << '\n'
              << "π/2 ≈ " << π / 2 << '\n'
              << "K(0.5) ≈ " << std::comp_ellint_1(0.5) << '\n'
              << "F(0.5, π/2) ≈ " << std::ellint_1(0.5, π / 2) << '\n'
              << "The period of a pendulum length 1 m at 10° initial angle ≈ "
              << 4 * std::sqrt(1 / 9.80665) * std::comp_ellint_1(std::sin(π / 18 / 2))
              << "s,\n" "whereas the linear approximation gives ≈ "
              << 2 * π * std::sqrt(1 / 9.80665) << '\n';
}

Output:

K(0) ≈ 1.5708
π/2 ≈ 1.5708
K(0.5) ≈ 1.68575
F(0.5, π/2) ≈ 1.68575
The period of a pendulum length 1 m at 10° initial angle ≈ 2.01024s,
whereas the linear approximation gives ≈ 2.00641

See also

ellint_1ellint_1fellint_1l (C++17)(C++17)(C++17)

(incomplete) elliptic integral of the first kind (function)

External links

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