std::sph_neumann, std::sph_neumannf, std::sph_neumannl

Since C++23

Header: <cmath>

1-3) Computes the spherical Bessel function of the second kind , also known as the spherical Neumann function, of n and x.The library provides overloads of std::sph_neumann for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)

# Declarations

float sph_neumann ( unsigned n, float x );
double sph_neumann ( unsigned n, double x );
long double sph_neumann ( unsigned n, long double x );

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

/* floating-point-type */ sph_neumann( unsigned n,
/* floating-point-type */ x );

(since C++23)

float sph_neumannf( unsigned n, float x );

(since C++17)

long double sph_neumannl( unsigned n, long double x );

(since C++17)

Additional overloads

template< class Integer >
double sph_neumann ( unsigned n, Integer x );

(since C++17)

# Parameters

# Return value

If no errors occur, returns the value of the spherical Bessel function of the second kind (spherical Neumann function) of n and x, that is nn(x) = (π/2x)1/2Nn+1/2(x) where Nn(x) is std::cyl_neumann (n, x) and x≥0.

# 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 argument num of integer type, std::sph_neumann(int_num, num) has the same effect as std::sph_neumann(int_num, static_cast(num)).

# Example

#include <cmath>
#include <iostream>
 
int main()
{
    // spot check for n == 1
    double x = 1.2345;
    std::cout << "n_1(" << x << ") = " << std::sph_neumann(1, x) << '\n';
 
    // exact solution for n_1
    std::cout << "-cos(x)/x² - sin(x)/x = "
              << -std::cos(x) / (x * x) - std::sin(x) / x << '\n';
}

# See also