ellint_fe

auto kyosu::ellint_fe = eve::functor<ellint_fe_t>

inlineconstexpr

Computes Jacobi's Amplitude function.

#include <kyosu/functions.hpp>

Callable Signatures

namespace kyosu
{
   // Regular overload
    constexpr Z       ellint_fe( real<Z>   z, real<U> m)               noexcept; //1
    constexpr auto    ellint_fe(complex<Z> z, real<U> m)               noexcept; //1
 
   //Semantic modifiers
   constexpr Z    ellint_fe[modular]( real<Z>      z, real<U> alpha)   noexcept; //1
   constexpr auto ellint_fe[modular](complex<Z>    z, real<U> alpha)   noexcept; //1
   constexpr Z    ellint_fe[eccentric]( real<Z>    z, real<U> m)       noexcept; //1
   constexpr auto ellint_fe[eccentric](complex<Z>  z, real<U> m)       noexcept; //1
   constexpr auto ellint_fe[threshold = tol]( real<Z>   z, real<U> k)  noexcept; //1
   constexpr auto ellint_fe[threshold = tol](complex<Z> z, real<U> k)  noexcept; //1
}

Parameters

• u: argument.
• m: amplitude parameter ( \(0\le m\le 1\)).
• alpha : modular angle in radian.
• k: elliptic modulus (eccentricity) .
• tol: accuracy tolerance (by defaut epsilon.
• c: Conditional expression masking the operation.
• l: Logical value masking the operation.

Return value

1. return the elliptic incomplete functions or first and second kind. Tafe care that the meaning of the second parameters depends on the option used (see note below).

Note

• \(\alpha\) is named the modular angle given in radian (modular option).
• \( k = \sin\alpha \) is named the elliptic modulus or eccentricity (eccentric option).
• \( m = k^2 = \sin^2\alpha\) is named the parameter (no option). Each of the above three quantities is completely determined by any of the others (given that they are non-negative). Thus, they can be used interchangeably up to roundings errors by giving the right option.

External references

• DLMF: Legendre Integrals
• Wolfram MathWorld: Jacobi Amplitude
• Wolfram MathWorld: Jacobi Amplitude
• Wikipedia: Elliptic integral

Example

#include <eve/wide.hpp>
#include <iomanip>
#include <iostream>
#include <kyosu/kyosu.hpp>
 
int main()
{
  std::cout << std::setprecision(16) << std::endl;
  using w_t = eve::wide<double, eve::fixed<4>>;
  auto z = kyosu::complex(w_t(2.0, 1.0, 1.0, 0.0), w_t(1.0, 1.0, 0.0, 1.0));
  auto m = w_t(0.5);
  auto [f, e] = kyosu::ellint_fe(z, eve::sqrt(m));
  std::cout << " z               " << z << std::endl;
  std::cout << " m               " << m << std::endl;
  std::cout << "f                " << f << std::endl;
  std::cout << "e                " << e << std::endl;
  return 0;
}