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

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Mathematical special functions
Defined in header <cmath>
(1)
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 ) ;
(2) (since C++17)
long double sph_neumannl ( unsigned n, long double x ) ;
(3) (since C++17)
Defined in header <cmath>
template < class Integer >
double sph_neumann ( unsigned n, Integer x ) ;
(A) (since C++17)
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)
A) Additional overloads are provided for all integer types, which are treated as double .

Parameters

n - the order of the function
x - the argument of the function

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 n n (x) = (π/2x) 1/2 N n+1/2 (x) where N n (x) is std:: cyl_neumann ( n, x ) and x≥0 .

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 n≥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 .

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 < double > ( num ) ) .

Example

Run this code 
#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';
}

Output: 

n_1(1.2345) = -0.981201
-cos(x)/x² - sin(x)/x = -0.981201

See also

(C++17) (C++17) (C++17)
cylindrical Neumann functions
(function)
(C++17) (C++17) (C++17)
spherical Bessel functions (of the first kind)
(function)

External links

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