agd

auto kyosu::agd = eve::functor<agd_t>

inlineconstexpr

Computes the principal branch inverse gudermanian of the argument.

Header file

#include <kyosu/functions.hpp>

Callable Signatures

namespace kyosu
{
  //  regular call
  constexpr auto agd(cayley_dickson_like z) noexcept;
  constexpr auto agd(cayley_dickson_like z, eve::value k)  noexcept;
 
  // semantic modifyers
  constexpr auto agd[real_only](Real z)                    noexcept;
  constexpr auto agd[radpi](cayley_dickson_like z)         noexcept;
  constexpr auto agd[rad](cayley_dickson_like z)           noexcept;
}

Parameters

• z: Value to process.

Return value

• A real typed input z is treated as if complex(z) was entered, unless the option real_only is used in which case the parameter must be a floating_value and the result will the same as a call to eve::agd,
• For complex input, returns \(\log(\tan(z/2 +\pi/4))\).
• For two parameters returns the kth branch of agd. If k is not a flint it is truncated before use.
• The radpi option provides a result expressed in \(\pi\) multiples.

Note

agd has branch cut discontinuities in \([\pi/2, 3\pi/2] + 2\pi n\) in the complex plane for integer \(n\). Morover, with real_only option the used formula is \(2*\mathbb{atan}(\tanh(z/2))\)

External references

• Wolfram MathWorld: Inverse Gudermannian
• Wikipedia: Gudermannian function
• Wikipedia: Inverse Gudermannian function

Example

#include <eve/wide.hpp>
#include <iostream>
#include <kyosu/kyosu.hpp>
 
int main()
{
  using kyosu::agd;
  using kyosu::complex_t;
  using kyosu::quaternion_t;
 
  std::cout << "Real:        ";
  std::cout << 72.9f << " -> " << agd(72.9f) << "\n";
 
  std::cout << "Complex:     ";
  std::cout << kyosu::complex_t<float>(3.5f, -2.9f) << " -> " << agd(kyosu::complex_t<float>(3.5f, -2.9f)) << "\n";
 
  std::cout << "Quaternion:  ";
  std::cout << kyosu::quaternion_t<double>(1., 2., 3., 4.) << " -> " << agd(kyosu::quaternion_t<double>(1., 2., 3., 4.))
            << "\n";
 
  std::cout << "SIMD:        ";
  using wc_t = eve::wide<kyosu::complex_t<double>, eve::fixed<2>>;
  std::cout << wc_t(kyosu::complex_t<double>(1.3, -3.7)) << " -> " << agd(wc_t(kyosu::complex_t<double>(1.3, -3.7)))
            << "\n";
 
  return 0;
}