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

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< cpp‎ | numeric‎ | 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)
Additional overloads |  |
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

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)

### External links

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

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