std::sph_neumann,std::sph_neumannf,std::sph_neumannl

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Numerics library

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C-style checked integer arithmetic

ckd_add (C++26)
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Mathematical special functions

assoc_laguerreassoc_laguerrefassoc_laguerrel
assoc_legendreassoc_legendrefassoc_legendrel
betabetafbetal
comp_ellint_1comp_ellint_1fcomp_ellint_1l
comp_ellint_2comp_ellint_2fcomp_ellint_2l
comp_ellint_3comp_ellint_3fcomp_ellint_3l
cyl_bessel_icyl_bessel_ifcyl_bessel_il

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl
cyl_bessel_kcyl_bessel_kfcyl_bessel_kl
cyl_neumanncyl_neumannfcyl_neumannl
ellint_1ellint_1fellint_1l
ellint_2ellint_2fellint_2l
ellint_3ellint_3fellint_3l
expintexpintfexpintl

hermitehermitefhermitel
laguerrelaguerreflaguerrel
legendrelegendreflegendrel
riemann_zetariemann_zetafriemann_zetal
sph_besselsph_besselfsph_bessell
sph_legendresph_legendrefsph_legendrel
sph_neumannsph_neumannfsph_neumannl

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Defined in header`<cmath>`
	(1)
float sph_neumann(unsigned n,float x); double sph_neumann(unsigned n,double x); longdouble sph_neumann(unsigned n,longdouble x);			(since C++17) (until C++23)
/* floating-point-type / sph_neumann(unsigned n, / floating-point-type */ x);			(since C++23)
float sph_neumannf(unsigned n,float x);		(2)	(since C++17)
longdouble sph_neumannl(unsigned n,longdouble x);		(3)	(since C++17)
Additional overloads
Defined in header`<cmath>`
template<class Integer> double sph_neumann(unsigned n, Integer x);		(A)	(since C++17)

1-3) Computes thespherical Bessel function of the second kind, also known as the spherical Neumann function, ofn andx. The library provides overloads ofstd::sph_neumann for all cv-unqualified floating-point types as the type of the parameterx.(since C++23)

A) Additional overloads are provided for all integer types, which are treated asdouble.

[edit]Parameters

n	-	the order of the function
x	-	the argument of the function

[edit]Return value

If no errors occur, returns the value of the spherical Bessel function of the second kind (spherical Neumann function) ofn andx, that isnn(x) = (π/2x)1/2
Nn+1/2(x) whereNn(x) isstd::cyl_neumann(n, x) andx≥0.

[edit]Error handling

Errors may be reported as specified inmath_errhandling

If the argument is NaN, NaN is returned and domain error is not reported
Ifn≥128, the behavior is implementation-defined

[edit]Notes

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 argumentnum of integer type,std::sph_neumann(int_num, num) has the same effect asstd::sph_neumann(int_num,static_cast<double>(num)).

[edit]Example

Run this code

#include <cmath>#include <iostream> int main(){// spot check for n == 1double x=1.2345;std::cout<<"n_1("<< x<<") = "<< std::sph_neumann(1, x)<<'\n'; // exact solution for n_1std::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

[edit]See also

cyl_neumanncyl_neumannfcyl_neumannl (C++17)(C++17)(C++17)	cylindrical Neumann functions (function)[edit]
sph_besselsph_besselfsph_bessell (C++17)(C++17)(C++17)	spherical Bessel functions (of the first kind) (function)[edit]