lambda

kyosu::lambda = eve::functor<lambda_t>

inlineconstexpr

Callable object computing The Dirichlet \( \displaystyle \lambda(z) = \sum_0^\infty \frac{1}{(2n+1)^z}\).

This function can be extended to the whole complex plane as \(\lambda(z) = \zeta(z)(1-2^{-z})\) (where \(\zeta\) is the Riemann zeta function). It coincides with the serie where the serie converges. However for z = 1 the result is \(\infty\). The usual extension mechanism is used for general Cayley-dickson input values.

Header file

#include <kyosu/functions.hpp>

Callable Signatures

namespace kyosu
{
   template<unsigned_scalar_value K, eve::ordered_value T>              constexpr auto lambda(T z) noexcept;  //1
   template<unsigned_scalar_value K, kyosu::concepts::cayley-dickson T> constexpr auto lambda(T z) noexcept;  //2
}

Parameters

• z : cayley_dickson or real value to process.

Return value

Returns the Dirichlet sum \( \displaystyle \sum_0^\infty \frac{1}{(2n+1)^z}\)

External references

• Wolfram MathWorld: Dirichlet lambda
• Wikipedia: Dirichlet series

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};
  wide_ft ref2 = { 0.0, 1.0, 2.0, 3.0};
  auto zc = kyosu::complex_t<wide_ft>(ref1, imf1);
 
  std::cout
    << "---- simd" << std::endl
    << "<- zc              = " << zc << std::endl
    << "<- ref2            = " << ref2 <<  std::endl
    << "-> lambda(zc)      = " << kyosu::lambda(zc)<< std::endl
    << "-> lambda(ref2)    = " << kyosu::lambda(ref2) << std::endl;
 
  return 0;
}