| Common mathematical functions | |||||||||||||||||||||||||||||||
| Mathematical special functions(C++17) | |||||||||||||||||||||||||||||||
| Mathematical constants(C++20) | |||||||||||||||||||||||||||||||
| Basic linear algebra algorithms(C++26) | |||||||||||||||||||||||||||||||
| Data-parallel types (SIMD)(C++26) | |||||||||||||||||||||||||||||||
| Floating-point environment(C++11) | |||||||||||||||||||||||||||||||
| Complex numbers | |||||||||||||||||||||||||||||||
Numeric array (valarray) | |||||||||||||||||||||||||||||||
| Pseudo-random number generation | |||||||||||||||||||||||||||||||
| Bit manipulation(C++20) | |||||||||||||||||||||||||||||||
| Saturation arithmetic(C++26) | |||||||||||||||||||||||||||||||
| Factor operations | |||||||||||||||||||||||||||||||
|
| ||||||||||||||||||||||||||||||
| Interpolations | |||||||||||||||||||||||||||||||
|
| ||||||||||||||||||||||||||||||
| Generic numeric operations | |||||||||||||||||||||||||||||||
| |||||||||||||||||||||||||||||||
| C-style checked integer arithmetic | |||||||||||||||||||||||||||||||
| |||||||||||||||||||||||||||||||
Defined in header <cmath> | ||
| (1) | ||
float cyl_neumann(float nu,float x); double cyl_neumann(double nu,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) |
longdouble cyl_neumannl(longdouble nu,longdouble x); | (3) | (since C++17) |
Defined in header <cmath> | ||
template<class Arithmetic1,class Arithmetic2> /* common-floating-point-type */ | (A) | (since C++17) |
std::cyl_neumann for all cv-unqualified floating-point types as the type of the parametersnu andx.(since C++23)Contents |
| nu | - | the order of the function |
| x | - | the argument of the function |
nu andx, is returned, that isNnu(x) =| Jnu(x)cos(nuπ)-J-nu(x) |
| sin(nuπ) |
Errors may be reported as specified inmath_errhandling:
Implementations that do not support C++17, but supportISO 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 headertr1/cmath and namespacestd::tr1.
An implementation of this function is also available inboost.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 argumentnum1 and second argumentnum2:
| (until C++23) |
Ifnum1 andnum2 have arithmetic types, thenstd::cyl_neumann(num1, num2) has the same effect asstd::cyl_neumann(static_cast</* common-floating-point-type */>(num1), If no such floating-point type with the greatest rank and subrank exists, thenoverload resolution does not result in a usable candidate from the overloads provided. | (since C++23) |
#include <cassert>#include <cmath>#include <iostream>#include <numbers> constdouble 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){returnstd::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){returnstd::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;constdouble nu=0.5;for(double x=0.0; x<=2.0; x+=0.333){constdouble n= std::cyl_neumann(nu, x);constdouble 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 == 0.5N_.5(+0.000000) = -inf, calculated via J = -infN_.5(+0.333000) = -1.306713, calculated via J = -1.306713N_.5(+0.666000) = -0.768760, calculated via J = -0.768760N_.5(+0.999000) = -0.431986, calculated via J = -0.431986N_.5(+1.332000) = -0.163524, calculated via J = -0.163524N_.5(+1.665000) = +0.058165, calculated via J = +0.058165N_.5(+1.998000) = +0.233876, calculated via J = +0.233876
(C++17)(C++17)(C++17) | regular modified cylindrical Bessel functions (function)[edit] |
(C++17)(C++17)(C++17) | cylindrical Bessel functions (of the first kind) (function)[edit] |
(C++17)(C++17)(C++17) | irregular modified cylindrical Bessel functions (function)[edit] |
| Weisstein, Eric W. "Bessel Function of the Second Kind." From MathWorld — A Wolfram Web Resource. |
