NAME

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

Synopsis

Defined in header <cmath>
double cyl_neumann( double ν, double x );

float cyl_neumannf( float ν, float x ); (1) (since C++17)

long double cyl_neumannl( long double ν, long double x );
Promoted cyl_neumann( Arithmetic ν, Arithmetic x ); (2) (since C++17)

1) Computes the cylindrical Neumann function (also known as Bessel function of the
second kind or Weber function) of ν and x.
2) A set of overloads or a function template for all combinations of arguments of
arithmetic type not covered by (1). If any argument has integral type, it is cast to
double. If any argument is long double, then the return type Promoted is also long
double, otherwise the return type is always double.

Parameters

ν - 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 ν and x, is returned, that is N
ν(x) =

J
ν(x)cos(νπ)-J
-ν(x)
sin(νπ)

(where J
ν(x) is std::cyl_bessel_j(ν,x)) for x≥0 and non-integer ν; for integer ν 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 ν>=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 <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 the J₋ᵥ, because the direct invocation of the
// std::cyl_bessel_j(ν,x), per formula above, for negative ν raises 'std::domain_error':
// Bad argument in __cyl_bessel_j.

double J₋ᵥ (double ν, double x) {
return std::cos(-ν*π) * std::cyl_bessel_j(-ν,x)
-std::sin(-ν*π) * std::cyl_neumann(-ν,x);
}

double J₊ᵥ (double ν, double x) { return std::cyl_bessel_j(ν,x); }

double Jᵥ (double ν, double x) { return ν < 0.0 ? J₋ᵥ(ν,x) : J₊ᵥ(ν,x); }

int main()
{
std::cout << "spot checks for ν == 0.5\n" << std::fixed << std::showpos;
double ν = 0.5;
for (double x = 0.0; x <= 2.0; x += 0.333) {
const double n = std::cyl_neumann(ν, x);
const double j = (Jᵥ(ν, x)*std::cos(ν*π) - Jᵥ(-ν, x)) / std::sin(ν*π);
std::cout << "N_.5(" << x << ") = " << n << ", calculated via J = " << j << '\n';
assert(n == j);
}
}

Output:

spot checks for ν == 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

External links

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