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

double      cyl_bessel_k( double ν, double x );

float       cyl_bessel_kf( float ν, float x  );

long double cyl_bessel_kl( long double ν, long double x );
(1) (since C++17)
Promoted    cyl_bessel_k( Arithmetic ν, Arithmetic x );
(2) (since C++17)
1) Computes the irregular modified cylindrical Bessel function (also known as modified Bessel function of the second kind) of ν and x.
2) A set of overloads or a function template for all combinations of arguments of arithmetic type not covered by (1). If any argument has integral type, it is cast to double. If any argument is long double, then the return type Promoted is also long double, otherwise the return type is always double.


ν - 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 ν and x, is returned, that is K
(x) =
(where I
is std::cyl_bessel_i(ν,x)) for x≥0 and non-integer ν; for integer ν 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 ν>=128, the behavior is implementation-defined


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


#include <cmath>
#include <iostream>
int main()
    double pi = std::acos(-1);
    double x = 1.2345;
    // spot check for ν == 0.5
    std::cout << "K_.5(" << x << ") = " << std::cyl_bessel_k( .5, x) << '\n'
              << "calculated via I = " << 
                     -std::cyl_bessel_i(.5,x))/std::sin(.5*pi) << '\n';


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

External links

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

See also

regular modified cylindrical Bessel functions
cylindrical Bessel functions (of the first kind)