std::cyl_neumann,std::cyl_neumannf,std::cyl_neumannl(3)

NAME
std::cyl_neumann,std::cyl_neumannf,std::cyl_neumannl - std::cyl_neumann,std::cyl_neumannf,std::cyl_neumannl

Synopsis
   Defined in header <cmath>
   double cyl_neumann( double ν, double x );
   float cyl_neumannf( float ν, float x ); (1) (since C++17)
   long double cyl_neumannl( long double ν, long double x );
   Promoted cyl_neumann( Arithmetic ν, Arithmetic x ); (2) (since C++17)
   1) Computes the cylindrical Neumann function (also known as Bessel function of the
   second kind or Weber function) 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.

Parameters
   ν - the order of the function
   x - the argument of the function

Return value
   If no errors occur, value of the cylindrical Neumann function (Bessel function of
   the second kind) of ν and x, is returned, that is N
   ν(x) =
   J
   ν(x)cos(νπ)-J
   -ν(x)
   sin(νπ)
   (where J
   ν(x) is std::cyl_bessel_j(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

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

Example
// Run this code
 #include <cassert>
 #include <cmath>
 #include <iostream>
 #include <numbers>
 const double π = std::numbers::pi; // or std::acos(-1) in pre C++20
 // To calculate the cylindrical Neumann function via cylindrical Bessel function of the
 // first kind we have to implement the J₋ᵥ, because the direct invocation of the
 // std::cyl_bessel_j(ν,x), per formula above, for negative ν raises 'std::domain_error':
 // Bad argument in __cyl_bessel_j.
 double J₋ᵥ (double ν, double x) {
     return std::cos(-ν*π) * std::cyl_bessel_j(-ν,x)
           -std::sin(-ν*π) * std::cyl_neumann(-ν,x);
 }
 double J₊ᵥ (double ν, double x) { return std::cyl_bessel_j(ν,x); }
 double Jᵥ (double ν, double x) { return ν < 0.0 ? J₋ᵥ(ν,x) : J₊ᵥ(ν,x); }
 int main()
 {
     std::cout << "spot checks for ν == 0.5\n" << std::fixed << std::showpos;
     double ν = 0.5;
     for (double x = 0.0; x <= 2.0; x += 0.333) {
         const double n = std::cyl_neumann(ν, x);
         const double j = (Jᵥ(ν, x)*std::cos(ν*π) - Jᵥ(-ν, x)) / std::sin(ν*π);
         std::cout << "N_.5(" << x << ") = " << n << ", calculated via J = " << j << '\n';
         assert(n == j);
     }
 }

Output:
 spot checks for ν == 0.5
 N_.5(+0.000000) = -inf, calculated via J = -inf
 N_.5(+0.333000) = -1.306713, calculated via J = -1.306713
 N_.5(+0.666000) = -0.768760, calculated via J = -0.768760
 N_.5(+0.999000) = -0.431986, calculated via J = -0.431986
 N_.5(+1.332000) = -0.163524, calculated via J = -0.163524
 N_.5(+1.665000) = +0.058165, calculated via J = +0.058165
 N_.5(+1.998000) = +0.233876, calculated via J = +0.233876

See also
   cyl_bessel_i
   cyl_bessel_if
   cyl_bessel_il regular modified cylindrical Bessel functions
   (C++17) (function)
   (C++17)
   (C++17)
   cyl_bessel_j
   cyl_bessel_jf
   cyl_bessel_jl cylindrical Bessel functions (of the first kind)
   (C++17) (function)
   (C++17)
   (C++17)
   cyl_bessel_k
   cyl_bessel_kf
   cyl_bessel_kl irregular modified cylindrical Bessel functions
   (C++17) (function)
   (C++17)
   (C++17)

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