cyl_bessel_j

kyosu::cyl_bessel_j = {}

inlineconstexpr

Computes the Bessel functions of the first kind, \( J_{\nu}(x)=\sum_{p=0}^{\infty}{\frac{(-1)^p}{p!\,\Gamma (p+\nu +1)}} {\left({x \over 2}\right)}^{2p+\nu }\) extended to the complex plane and cayley_dickson values.

It is the solution of \( x^{2}y''+xy'+(x^2-\nu^2)y=0\) for which \( y(0) = 0\) if \(\nu \ne 0\) else \(1\).

#include <kyosu/functions.hpp>

Callable Signatures

namespace kyosu
{
   template<eve::floating_ordered_value, kyosu::concepts::cayley_dickson T>
   constexpr auto cyl_bessel_j(N nu, T z) noexcept;
 
   template<eve::floating_ordered_value, eve::floating_ordered_value T>
   constexpr T    cyl_bessel_j(N n, T z) noexcept;
}

Parameters

• nu: order of the function.
• z: Value to process.

Return value

• returns \(J_\nu(z)\).

Example

#include <kyosu/kyosu.hpp>
#include <eve/wide.hpp>
#include <iostream>
 
int main()
{
  std::cout.precision(16);
  using w_t = eve::wide<double, eve::fixed<2>>;
  auto z = kyosu::complex(w_t(20.0, 1.5), w_t(0.0, 1.5));
  auto v = -(2+1/3.0);
  int nb = int(eve::abs(v)+1);
  std::cout << "z                    " << z << std::endl;
  std::vector<decltype(z)> js(nb);
  kyosu::cyl_bessel_j(v, z, js);
  auto inc = eve::sign(v);
  auto v0 =  eve::frac(v);
  for(int n=0; n < nb; ++n)
  {
   std::cout << "js[" << n << "]                    = " << js[n] << std::endl;
    std::cout << "j(" << v0 << ",  z) " << kyosu::cyl_bessel_j(v0, z) << std::endl;
    v0 += inc;
  }
  return 0;
}