ellint_d

auto eve::ellint_d = functor<ellint_d_t>

inlineconstexpr

Header file

#include <eve/module/elliptic.hpp>

Callable Signatures

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

Parameters

• phi: floating Jacobi amplitude.
• k: floating elliptic modulus. k must satisfy \(k^2\sin^2\phi \le 1\) or the result is NaN. In the complete case this means \(|k| \le 1\).
• c: Conditional expression masking the operation.
• m: Logical value masking the operation.

Return value

1. the complete elliptic integral ( corresponding to \( \phi = \pi/2 \) ): \( \mathbf{D}(k) = \int_0^{\pi/2} \frac{\sin^2 t}{\sqrt{1-k^2\sin^2 t}} \scriptstyle\;\mathrm{d}t\) is returned.
2. the incomplete elliptic integral of the first kind: \( \mathbf{D}(k) = \int_0^{\pi/2} \frac{\sin^2 t}{\sqrt{1-k^2\sin^2 t}} \scriptstyle\;\mathrm{d}t\) is returned:

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

• 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_d(wf)                = " << eve::ellint_d(wf) << "\n";
  std::cout << "-> ellint_d(wphi, wf)          = " << eve::ellint_d(wphi, wf) << "\n";
  std::cout << "-> ellint_d[ignore_last(2)](wf)= " << eve::ellint_d[eve::ignore_last(2)](wf) << "\n";
  std::cout << "-> ellint_d[wf != 1.0](wf)     = " << eve::ellint_d[wf != -1.0f](wf) << "\n";
  std::cout << "-> ellint_d(pio_2[lower],wf)   = " << eve::ellint_d(eve::pio_2[eve::lower](eve::as(wphi)), wf) << '\n';
}