xi

kyosu::xi = eve::functor<xi_t>

inlineconstexpr

Computes the Riemann \( \displaystyle\xi(z) = \frac{1}{2}z(z-1)\pi^{-\frac{z}{2}}\Gamma(\frac{z}{2})\zeta(z)\). function or the Landau version \( \displaystyle\Xi(z) = \xi(\frac{1}{2} + i z)\).

Header file

#include <kyosu/functions.hpp>

Callable Signatures

namespace kyosu
{
   //regular call
   constexpr auto xi(auto z) noexcept;           // 1
 
   // Semantic modifyiers
   constexpr auto xi[riemann](auto z) noexcept;  // 1
   constexpr auto xi[landau](T z) noexcept;      // 2
}

Parameters

• z : cayley-dickson like value to process.

Return value

1. Returns the value at z of the Riemann \(\xi\) function (ξ can be used a an alias).
2. Returns the value at z of the Landau \(\Xi\) function (Ξ can be used a an alias).

if the input z is a floating_value the call is done as if complex(z) was input.

External references

• Wikipedia: Riemann xi function

Example

#include <kyosu/kyosu.hpp>
#include <eve/wide.hpp>
#include <iostream>
 
int main()
{
  using wide_ft = eve::wide <float, eve::fixed<4>>;
  wide_ft ref1 = { 3.0f, 2.0f, 1.0f, 0.5f};
  wide_ft imf1 = { 2.0f , -1.0,  -5.0, 0.0};
  auto zc = kyosu::complex_t<wide_ft>(ref1, imf1);
  std::cout
    << "---- simd" << std::endl
    << "<- zc              = " << zc << std::endl
    << "<- ref1            = " << ref1 << std::endl
    << "-> xi(zc)          = " << kyosu::xi(zc) << std::endl
    << "-> xi[riemann](zc) = " << kyosu::xi[kyosu::riemann](zc) << std::endl
    << "-> xi[landau ](zc) = " << kyosu::xi[kyosu::landau] (zc) << std::endl
    << "-> ξ(zc)           = " << kyosu::ξ(zc)             << std::endl
    << "-> Ξ(zc)           = " << kyosu::Ξ(zc)                 << std::endl
    << "-> xi(ref1)        = " << kyosu::xi(ref1) << std::endl;
   return 0;
}