cyl_bessel_jn

kyosu::cyl_bessel_jn = {}

inlineconstexpr

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

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

#include <kyosu/functions.hpp>

Callable Signatures

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

Parameters

• z: Value to process.

Return value

• returns \(J_n(z)\).

Example

#include <kyosu/kyosu.hpp>
#include <eve/wide.hpp>
#include <iostream>
 
int main()
{
  size_t n = 3;
  using kyosu::cyl_bessel_jn;
  std::cout.precision(16);
  std::cout << std::scientific << std::endl;
  std::cout << "1.0             " << " -> " << cyl_bessel_jn(n,1.0) << "\n";
  std::cout << "1+0i            " << " -> " << cyl_bessel_jn(n,kyosu::complex(1.0, 0.0)) << "\n";
  std::cout << "15.0            " << " -> " << cyl_bessel_jn(n,15.0) << "\n";
  std::cout << "15+0i           " << " -> " << cyl_bessel_jn(n,kyosu::complex(15.0, 0.0)) << "\n";
  std::cout << "40.0            " << " -> " << cyl_bessel_jn(n,40.0) << "\n";
  std::cout << "40+0i           " << " -> " << cyl_bessel_jn(n,kyosu::complex(40.0, 0.0)) << "\n";
  std::cout << "60.0            " << " -> " << cyl_bessel_jn(n,60.0) << "\n";
  std::cout << "60+0i           " << " -> " << cyl_bessel_jn(n,kyosu::complex(60.0, 0.0)) << "\n";
 
  eve::wide<double, eve::fixed<4>> z(1.0, 15.0, 40.0, 60.0);
  auto zz = kyosu::complex(z);
  std::cout << z  << "\n -> " << cyl_bessel_jn(n,z) << "\n";
  std::cout << zz << "\n -> " << kyosu::real(cyl_bessel_jn(n,zz)) << "\n";
 
  std::cout << "1.0                 " << " -> " << cyl_bessel_jn(n,1.0) << "\n";
  std::cout << "1.0+36.0i           " << " -> " << cyl_bessel_jn(n,kyosu::complex(1.0, 36.0)) << "\n";
  std::cout << "1.0+36.0i+2.0j+1.5k " << " -> " << cyl_bessel_jn(n,kyosu::quaternion(1.0, 36.0, 2.0, 1.5)) << "\n";
 
 
  eve::wide<double, eve::fixed<4>> z1(1.0, 2.0, 40.0, 0.0),  z2(36.0, 0.5, 0.0, 40.0);
  auto z0 = kyosu::complex(z1, z2);
  std::cout << z0 << " \n-> " << cyl_bessel_jn(n,z0) << "\n";
  return 0;
}