std::cyl_neumann, std::cyl_neumannf, std::cyl_neumannl

Since C++23

Header: <cmath>

1-3) Computes the cylindrical Neumann function (also known as Bessel function of the second kind or Weber function) of nu and x.The library provides overloads of std::cyl_neumann for all cv-unqualified floating-point types as the type of the parameters nu and x.(since C++23)

# Declarations

float cyl_neumann ( float nu, float x );
double cyl_neumann ( double nu, double x );
long double cyl_neumann ( long double nu, long double x );

(since C++17) (until C++23)

/* floating-point-type */ cyl_neumann( /* floating-point-type */ nu,
/* floating-point-type */ x );

(since C++23)

float cyl_neumannf( float nu, float x );

(since C++17)

long double cyl_neumannl( long double nu, long double x );

(since C++17)

Additional overloads

template< class Arithmetic1, class Arithmetic2 >
/* common-floating-point-type */
cyl_neumann( Arithmetic1 nu, Arithmetic2 x );

(since C++17)

# Parameters

# 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 and num2 have arithmetic types, then std::cyl_neumann(num1, num2) has the same effect as std::cyl_neumann(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.

# Example

#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 J, because the direct invocation of the
// std::cyl_bessel_j(nu, x), per formula above,
// for negative nu raises 'std::domain_error': Bad argument in __cyl_bessel_j.
 
double J_neg(double nu, double x)
{
    return std::cos(-nu * π) * std::cyl_bessel_j(-nu, x)
          -std::sin(-nu * π) * std::cyl_neumann(-nu, x);
}
 
double J_pos(double nu, double x)
{
    return std::cyl_bessel_j(nu, x);
}
 
double J(double nu, double x)
{
    return nu < 0.0 ? J_neg(nu, x) : J_pos(nu, x);
}
 
int main()
{
    std::cout << "spot checks for nu == 0.5\n" << std::fixed << std::showpos;
    const double nu = 0.5;
    for (double x = 0.0; x <= 2.0; x += 0.333)
    {
        const double n = std::cyl_neumann(nu, x);
        const double j = (J(nu, x) * std::cos(nu * π) - J(-nu, x)) / std::sin(nu * π);
        std::cout << "N_.5(" << x << ") = " << n << ", calculated via J = " << j << '\n';
        assert(n == j);
    }
}

# See also