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

/* common-floating-point-type */

    cyl_neumann( Arithmetic1 nu, Arithmetic2 x );
 (A) (since 哋它亢++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 哋它亢++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 哋它亢++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 哋它亢++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 哋它亢++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 哋它亢++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

(哋它亢++17)(哋它亢++17)(哋它亢++17)
 regular modified cylindrical Bessel functions
(function)
(哋它亢++17)(哋它亢++17)(哋它亢++17)
 cylindrical Bessel functions (of the first kind)
(function)
(哋它亢++17)(哋它亢++17)(哋它亢++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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