ellint_2

auto eve::ellint_2 = functor<ellint_2_t>

inlineconstexpr

Header file

#include <eve/module/elliptic.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   constexpr auto ellint_2(floating_value auto k)                                      noexcept; // 1
   constexpr auto ellint_2(floating_value auto phi, floating_value auto k)             noexcept; // 2
   constexpr auto ellint_2[modular](floating_value auto phi, floating_value auto alpha)noexcept; // 2
   constexpr auto ellint_2[eccentric](floating_value auto phi, floating_value auto m)  noexcept; // 2
 
   // Lanes masking
   constexpr auto ellint_2[conditional_expr auto c](/*any of the above overloads*/)    noexcept; // 3
   constexpr auto ellint_2[logical_value auto l](/*any of the above overloads*/)       noexcept; // 3
}

Parameters

• phi: floating Jacobi amplitude.
• k: amplitude parameter ( \(0\le k\le 1\)).
• alpha: amodular angle given in radian (modular option).
• m : elliptic modulus or eccentricity (eccentric option).
• c: Conditional expression masking the operation.
• l: Logical value masking the operation.

Return value

1. the complete ( corresponding to \( \phi = \pi/2 \) )of the second kind: \(\mathbf{E}(k) = \int_0^{\pi/2} \scriptstyle \sqrt{1-k\sin^2 t}\,\mathrm{d}t\) is returned.
2. the incomplete elliptic integrals of the second kind: \( \mathbf{E}(\phi, k) = \int_0^{\phi} \scriptstyle \sqrt{1-k\sin^2 t}\,\mathrm{d}t\) is returned. \(\alpha\) is \(\sin k\) and \(m\) is \(\sqrt(k)\)
3. The operation is performed conditionally

Note

Be aware that as \(\pi/2\) is not exactly represented by floating point values the result of the incomplete function with a \(\phi\) floating point value representing \(\pi/2\) can differ a lot with the result of the complete call.

External references

• C++ standard reference: ellint_2
• DLMF: Elliptic Integrals
• Wolfram MathWorld: Elliptic Integral

Example

// revision 1
#include <eve/module/elliptic.hpp>
#include <iostream>
 
eve::wide wf{1.0, 0.0, 0.75, 0.5};
eve::wide wphi{1.0, 1.0e-30, 0.5, 0.0};
 
int main(){
   std::cout << "<- wf   = " << wf << "\n";
   std::cout << "<- wphi = " << wphi << "\n";
 
   std::cout << "-> ellint_2(wf)                = " << eve::ellint_2(wf) << "\n";
   std::cout << "-> ellint_2(wphi, wf)          = " << eve::ellint_2(wphi, wf) << "\n";
   std::cout << "-> ellint_2[ignore_last(2)](wf)= " << eve::ellint_2[eve::ignore_last(2)](wf) << "\n";
   std::cout << "-> ellint_2[wf != 1.0](wf)     = " << eve::ellint_2[wf != -1.0f](wf) << "\n";
   std::cout << "-> ellint_2(pio_2,wf)          = " << eve::ellint_2(eve::pio_2[eve::lower](eve::as(wphi)), wf) << '\n';
}