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/C++

std::comp_ellint_2, std::comp_ellint_2f, std::comp_ellint_2l

double      comp_ellint_2( double k);
float       comp_ellint_2( float k );
long double comp_ellint_2( long double k );
float       comp_ellint_2f( float k );
long double comp_ellint_2l( long double k );

(1)

(since C++17)

double      comp_ellint_2( IntegralType k );

(2)

(since C++17)

1) Computes the complete elliptic integral of the second 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

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 second kind of k, that is ellint_2(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 perimeter of an ellipse with eccentricity k and semimajor axis a equals 4aE(k), where E is std::comp_ellint_2. When eccentricity equals 0, the ellipse degenerates to a circle with radius a and the perimeter equals 2πa, so E(0) = π/2. When eccentricity equals 1, the ellipse degenerates to a line of length 2a, whose perimeter is 4a, so E(1) = 1.

Example

#include <cmath>
#include <iostream>
int main()
{
    double hpi = std::acos(-1)/2;
    std::cout << "E(0) = " << std::comp_ellint_2(0) << '\n'
              << "π/2 = " << hpi << '\n'
              << "E(1) = " << std::comp_ellint_2(1) << '\n'
              << "E(1, π/2) = " << std::ellint_2(1, hpi) << '\n';
}

Output:

E(0) = 1.5708
π/2 = 1.5708
E(1) = 1
E(1, π/2) = 1

External links

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