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

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< cpp‎ | numeric‎ | special math

double sph_neumann ( unsigned n, double x ); float sph_neumann ( unsigned n, float x ); long double sph_neumann ( unsigned n, long double x ); float sph_neumannf( unsigned n, float x ); long double sph_neumannl( unsigned n, long double x ); | (1) | (since C++17)
double sph_neumann( unsigned n, IntegralType x ); | (2) | (since C++17)

1) Computes the spherical Bessel function of the second kind, also known as the spherical Neumann function, of `n` and `x`.
2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to 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

### 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)/x^2 - (sin x)/x = "
                  << -std::cos(x)/(x*x) - std::sin(x)/x << '\n';
    }

Output:

    n_1(1.2345) = -0.981201
    -(cos x)/x^2 - (sin x)/x = -0.981201

### External links

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

### See also

cyl_neumann cyl_neumannf cyl_neumannl (C++17)(C++17)(C++17) | cylindrical Neumann functions (function)
sph_bessel sph_besself sph_bessell (C++17)(C++17)(C++17) | spherical Bessel functions (of the first kind) (function)

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