std:: cyl_bessel_k, std:: cyl_bessel_kf, std:: cyl_bessel_kl

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Defined in header <cmath>
(1)
float cyl_bessel_k ( float nu, float x ) ;

double cyl_bessel_k ( double nu, double x ) ;

long double cyl_bessel_k ( long double nu, long double x ) ;
(since C++17)
(until C++23)
/* floating-point-type */ cyl_bessel_k ( /* floating-point-type */ nu,
/* floating-point-type */ x ) ;
(since C++23)
float cyl_bessel_kf ( float nu, float x ) ;
(2) (since C++17)
long double cyl_bessel_kl ( long double nu, long double x ) ;
(3) (since C++17)
Defined in header <cmath>
template < class Arithmetic1, class Arithmetic2 >

/* common-floating-point-type */

cyl_bessel_k ( Arithmetic1 nu, Arithmetic2 x ) ;
(A) (since C++17)
1-3) Computes the irregular modified cylindrical Bessel function (also known as modified Bessel function of the second kind) of nu and x . The library provides overloads of std::cyl_bessel_k for all cv-unqualified floating-point types as the type of the parameters nu and x . (since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

Parameters

nu - the order of the function
x - the argument of the function

Return value

If no errors occur, value of the irregular modified cylindrical Bessel function (modified Bessel function of the second kind) of nu and x , is returned, that is K nu (x) =
π
2
I -nu (x)-I nu (x)
sin(nuπ)
(where I nu (x) is std:: cyl_bessel_i ( nu, x ) ) for x≥0 and non-integer nu ; for integer nu a limit is used.

Error handling

Errors may be reported as specified in math_errhandling :

  • If the argument is NaN, NaN is returned and domain error is not reported.
  • If nu≥128 , the behavior is implementation-defined.

Notes

Implementations that do not support C++17, but support ISO 29124:2010 , provide this function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header tr1/cmath and namespace std::tr1 .

An implementation of this function is also available in boost.math .

The additional overloads are not required to be provided exactly as (A) . They only need to be sufficient to ensure that for their first argument num1 and second argument num2 :

  • If num1 or num2 has type long double , then std :: cyl_bessel_k ( num1, num2 ) has the same effect as std :: cyl_bessel_k ( static_cast < long double > ( num1 ) ,
    static_cast < long double > ( num2 ) )     .
  • Otherwise, if num1 and/or num2 has type double or an integer type, then std :: cyl_bessel_k ( num1, num2 ) has the same effect as std :: cyl_bessel_k ( static_cast < double > ( num1 ) ,
    static_cast < double > ( num2 ) )     .
  • Otherwise, if num1 or num2 has type float , then std :: cyl_bessel_k ( num1, num2 ) has the same effect as std :: cyl_bessel_k ( static_cast < float > ( num1 ) ,
    static_cast < float > ( num2 ) )     .
(until C++23)

If num1 and num2 have arithmetic types, then std :: cyl_bessel_k ( num1, num2 ) has the same effect as std :: cyl_bessel_k ( static_cast < /* common-floating-point-type */ > ( num1 ) ,
static_cast < /* common-floating-point-type */ > ( num2 ) ) , where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2 , arguments of integer type are considered to have the same floating-point conversion rank as double .

If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided.

(since C++23)

Example

Run this code 
#include <cmath>
#include <iostream>
#include <numbers>
 
int main()
{
    double pi = std::numbers::pi;
    const double x = 1.2345;
 
    // spot check for nu == 0.5
    std::cout << "K_.5(" << x << ") = " << std::cyl_bessel_k(.5, x) << '\n'
              << "calculated via I = "
              << (pi / 2) * (std::cyl_bessel_i(-.5, x)
                 - std::cyl_bessel_i(.5, x)) / std::sin(.5 * pi) << '\n';
}

Output: 

K_.5(1.2345) = 0.32823
calculated via I = 0.32823

See also

(C++17) (C++17) (C++17)
regular modified cylindrical Bessel functions
(function)
(C++17) (C++17) (C++17)
cylindrical Bessel functions (of the first kind)
(function)

External links

Weisstein, Eric W. "Modified Bessel Function of the Second Kind." From MathWorld — A Wolfram Web Resource.
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