jacobi_elliptic

kyosu::jacobi_elliptic = eve::functor<jacobi_elliptic_t>

inlineconstexpr

Computes Jacobi's Amplitude function.

#include <kyosu/functions.hpp>

Callable Signatures

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

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

• returns the jacobian amplitude function. Take care that the meaning of the second parameters depends on the option(s) used (see note below).
• The threshold option can be used to choose an accuracy tolerance.

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

• C++ standard reference: am
• DLMF: Jacobi Amplitude
• Wolfram MathWorld: Jacobi Amplitude
• Wikipedia: Jacobi elliptic functions

Example

#include <kyosu/kyosu.hpp>
#include <eve/wide.hpp>
#include <iostream>
#include <iomanip>
 
int main()
{
  std::cout<< std::setprecision(16) << std::endl;
  using w_t = eve::wide<double, eve::fixed<2>>;
  auto z = kyosu::complex(w_t(2.0, 1.0), w_t(0.0, 1.0));
  auto m = w_t(0.5, 0.7);
  auto [sn, cn, dn] =  kyosu::jacobi_elliptic(z, m);
  std::cout << " z               " << z  << std::endl;
  std::cout << " m               " << m  << std::endl;
  std::cout << "sn(z, m)         " << sn << std::endl;
  std::cout << "cn(z, m)         " << cn << std::endl;
  std::cout << "dn(z, m)         " << dn << std::endl;
  return 0;
}