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