cyl_bessel_kn

auto eve::cyl_bessel_kn = functor<cyl_bessel_kn_t>

inlineconstexpr

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

Header file

#include <eve/module/bessel.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   template<value N, floating_value T> constexpr as_wide_as_t<T,N> cyl_bessel_kn(N n, T x)    noexcept; // 1
 
   // Lanes masking
   constexpr auto cyl_bessel_kn[conditional_expr auto c](value auto n, floating_value auto x) noexcept; // 2
   constexpr auto cyl_bessel_kn[logical_value auto m](value auto n, floating_value auto x)    noexcept; // 2
}

Parameters

• n: order of the function (non necessarily integral)
• x: floating argument.
• c: Conditional expression masking the operation.
• m: Logical value masking the operation.

Return value

1. The value of \(\displaystyle K_n(x)=\frac{\Gamma(n+1/2)(2x)^n}{\sqrt\pi} \int_{0}^{\infty}\frac{\cos\tau}{(\tau^2+x^2)^{n+1/2}}\,\mathrm{d}\tau\) is returned.
2. The operation is performed conditionnaly.

External references

• C++ standard reference: cyl_bessel_k
• Wikipedia: Bessel Functions
• DLMF: Modified Bessel Functions

Example

// revision 1
#include <eve/module/bessel.hpp>
#include <iostream>
 
 
eve::wide<double> wf([](auto i, auto c)->double{ return 2*(i+c/2);});
eve::wide<std::uint64_t> wu([](auto i, auto )->std::uint64_t{ return 2*i;});
eve::wide w{0.5, 1.5, 0.1, 1.0, 19.0, 25.0, 21.5, 10000.0};
eve::wide m{0.5, -1.0, 1.5, -2.0, 2.5, -2.6, 3.2, -12.0};
 
int main()
{
   std::cout << "<- wf = " << wf << "\n";
   std::cout << "<- wu = " << wu << "\n";
   std::cout << "<- m  = " << m << "\n";
   std::cout << "<- w  = " << w << "\n";
 
   std::cout << "-> cyl_bessel_kn(wu, wf)                = " << eve::cyl_bessel_kn(wu, wf) << "\n";
   std::cout << "-> cyl_bessel_kn[ignore_last(2)](wu, wf)= " << eve::cyl_bessel_kn[eve::ignore_last(2)](wu, wf) << "\n";
   std::cout << "-> cyl_bessel_kn[wu != 2u](wu, wf)      = " << eve::cyl_bessel_kn[wu != 2u](wu, wf) << "\n";
   std::cout << "-> cyl_bessel_kn(m, w)                  = " << eve::cyl_bessel_kn(m, w) << "\n";
}