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/C++

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

double      ellint_3( double k, double ν, double φ );
float       ellint_3f( float k, float ν, float φ  );
long double ellint_3l( long double k, long double ν, long double φ );

(1)

(since C++17)

Promoted    ellint_3( Arithmetic k, Arithmetic ν, Arithmetic φ );

(2)

(since C++17)

1) Computes the incomplete elliptic integral of the third kind of k, ν, and φ.

2) A set of overloads or a function template for all combinations of arguments of arithmetic type not covered by (1). If any argument has integral type, it is cast to double. If any argument is long double, then the return type Promoted is also long double, otherwise the return type is always double.

Parameters

k	-	elliptic modulus or eccentricity (a value of a floating-point or integral type)
ν	-	elliptic characteristic (a value of floating-point or integral type)
φ	-	Jacobi amplitude (a value of floating-point or integral type, measured in radians)

Return value

If no errors occur, value of the incomplete elliptic integral of the third kind of k, ν, and φ, that is ∫φ
0

dθ

(1-νsin2
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.

Example

#include <cmath>
#include <iostream>
int main()
{
    double hpi = std::acos(-1)/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

External links

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