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

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Defined in header <cmath>
(1)
double comp_ellint_1 ( double k ) ;

float comp_ellint_1 ( float k ) ;

long double comp_ellint_1 ( long double k ) ;
(since C++17)
(until C++23)
/* floating-point-type */ comp_ellint_1 ( /* floating-point-type */ k ) ;
(since C++23)
float comp_ellint_1f ( float k ) ;
(2) (since C++17)
long double comp_ellint_1l ( long double k ) ;
(3) (since C++17)
Defined in header <cmath>
template < class Integer >
double comp_ellint_1 ( Integer k ) ;
(A) (since C++17)
1-3) Computes the complete elliptic integral of the first kind of k . The library provides overloads of std::comp_ellint_1 for all cv-unqualified floating-point types as the type of the parameter k . (since C++23)
A) Additional overloads are provided for all integer types, which are treated as double .

Parameters

k - elliptic modulus or eccentricity (a floating-point or integer value)

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 .

The additional overloads are not required to be provided exactly as (A) . They only need to be sufficient to ensure that for their argument num of integer type, std :: comp_ellint_1 ( num ) has the same effect as std :: comp_ellint_1 ( static_cast < double > ( num ) ) .

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 1m 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

(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.
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