std:: cyl_neumann, std:: cyl_neumannf, std:: cyl_neumannl

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Mathematical special functions
Defined in header <cmath>
(1)
float cyl_neumann ( float nu, float x ) ;

double cyl_neumann ( double nu, double x ) ;

long double cyl_neumann ( long double nu, long double x ) ;
(since C++17)
(until C++23)
/* floating-point-type */ cyl_neumann ( /* floating-point-type */ nu,
/* floating-point-type */ x ) ;
(since C++23)
float cyl_neumannf ( float nu, float x ) ;
(2) (since C++17)
long double cyl_neumannl ( long double nu, long double x ) ;
(3) (since C++17)
Defined in header <cmath>
template < class Arithmetic1, class Arithmetic2 >

/* common-floating-point-type */

cyl_neumann ( Arithmetic1 nu, Arithmetic2 x ) ;
(A) (since C++17)
1-3) Computes the cylindrical Neumann function (also known as Bessel function of the second kind or Weber function) of nu and x . The library provides overloads of std::cyl_neumann for all cv-unqualified floating-point types as the type of the parameters nu and x . (since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

Parameters

nu - the order of the function
x - the argument of the function

Return value

If no errors occur, value of the cylindrical Neumann function (Bessel function of the second kind) of nu and x , is returned, that is N nu (x) =
J nu (x)cos(nuπ)-J -nu (x)
sin(nuπ)
(where J nu (x) is std:: cyl_bessel_j ( nu, x ) ) for x≥0 and non-integer nu ; for integer nu a limit is used.

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 nu≥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 first argument num1 and second argument num2 :

  • If num1 or num2 has type long double , then std :: cyl_neumann ( num1, num2 ) has the same effect as std :: cyl_neumann ( static_cast < long double > ( num1 ) ,
    static_cast < long double > ( num2 ) )     .
  • Otherwise, if num1 and/or num2 has type double or an integer type, then std :: cyl_neumann ( num1, num2 ) has the same effect as std :: cyl_neumann ( static_cast < double > ( num1 ) ,
    static_cast < double > ( num2 ) )     .
  • Otherwise, if num1 or num2 has type float , then std :: cyl_neumann ( num1, num2 ) has the same effect as std :: cyl_neumann ( static_cast < float > ( num1 ) ,
    static_cast < float > ( num2 ) )     .
(until C++23)

If num1 and num2 have arithmetic types, then std :: cyl_neumann ( num1, num2 ) has the same effect as std :: cyl_neumann ( static_cast < /* common-floating-point-type */ > ( num1 ) ,
static_cast < /* common-floating-point-type */ > ( num2 ) ) , where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2 , arguments of integer type are considered to have the same floating-point conversion rank as double .

If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided.

(since C++23)

Example

Run this code 
#include <cassert>
#include <cmath>
#include <iostream>
#include <numbers>
 
const double π = std::numbers::pi; // or std::acos(-1) in pre C++20
 
// To calculate the cylindrical Neumann function via cylindrical Bessel function of the
// first kind we have to implement J, because the direct invocation of the
// std::cyl_bessel_j(nu, x), per formula above,
// for negative nu raises 'std::domain_error': Bad argument in __cyl_bessel_j.
 
double J_neg(double nu, double x)
{
    return std::cos(-nu * π) * std::cyl_bessel_j(-nu, x)
          -std::sin(-nu * π) * std::cyl_neumann(-nu, x);
}
 
double J_pos(double nu, double x)
{
    return std::cyl_bessel_j(nu, x);
}
 
double J(double nu, double x)
{
    return nu < 0.0 ? J_neg(nu, x) : J_pos(nu, x);
}
 
int main()
{
    std::cout << "spot checks for nu == 0.5\n" << std::fixed << std::showpos;
    const double nu = 0.5;
    for (double x = 0.0; x <= 2.0; x += 0.333)
    {
        const double n = std::cyl_neumann(nu, x);
        const double j = (J(nu, x) * std::cos(nu * π) - J(-nu, x)) / std::sin(nu * π);
        std::cout << "N_.5(" << x << ") = " << n << ", calculated via J = " << j << '\n';
        assert(n == j);
    }
}

Output: 

spot checks for nu == 0.5
N_.5(+0.000000) = -inf, calculated via J = -inf
N_.5(+0.333000) = -1.306713, calculated via J = -1.306713
N_.5(+0.666000) = -0.768760, calculated via J = -0.768760
N_.5(+0.999000) = -0.431986, calculated via J = -0.431986
N_.5(+1.332000) = -0.163524, calculated via J = -0.163524
N_.5(+1.665000) = +0.058165, calculated via J = +0.058165
N_.5(+1.998000) = +0.233876, calculated via J = +0.233876

See also

(C++17) (C++17) (C++17)
regular modified cylindrical Bessel functions
(function)
(C++17) (C++17) (C++17)
cylindrical Bessel functions (of the first kind)
(function)
(C++17) (C++17) (C++17)
irregular modified cylindrical Bessel functions
(function)

External links

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