std::sph_neumann,std::sph_neumannf,std::sph_neumannl(3)

NAME
std::sph_neumann,std::sph_neumannf,std::sph_neumannl - std::sph_neumann,std::sph_neumannf,std::sph_neumannl

Synopsis
   Defined in header <cmath>
   float sph_neumann ( unsigned n, float x );
                        (since C++17)
   double sph_neumann ( unsigned n, double x ); (until C++23)
   long double sph_neumann ( unsigned n, long double x );
   /* floating-point-type */ sph_neumann( unsigned n,
                        /* floating-point-type (1) (since C++23)
   */ x );
   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 > (A) (since C++17)
   double sph_neumann ( unsigned n, Integer x );
   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 cylindrical Neumann functions
   (C++17) (function)
   (C++17)
   (C++17)
   sph_bessel
   sph_besself
   sph_bessell spherical Bessel functions (of the first kind)
   (C++17) (function)
   (C++17)
   (C++17)

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