std::sph_legendre, std::sph_legendref, std::sph_legendrel

 

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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

 

Defined in header <cmath>
	(1)
float       sph_legendre ( unsigned l, unsigned m, float theta ); double      sph_legendre ( unsigned l, unsigned m, double theta ); long double sph_legendre ( unsigned l, unsigned m, long double theta );			(since C++17) (until C++23)
/* floating-point-type */ sph_legendre( unsigned l, unsigned m,                                         /* floating-point-type */ theta );			(since C++23)
float       sph_legendref( unsigned l, unsigned m, float theta );		(2)	(since C++17)
long double sph_legendrel( unsigned l, unsigned m, long double theta );		(3)	(since C++17)
Additional overloads
Defined in header <cmath>
template< class Integer > double      sph_legendre ( unsigned l, unsigned m, Integer theta );		(A)	(since C++17)

1-3) Computes the spherical associated Legendre function of degree l, order m, and polar angle theta. The library provides overloads of std::sph_legendre for all cv-unqualified floating-point types as the type of the parameter theta. (since C++23)

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

Parameters

l	-	degree
m	-	order
theta	-	polar angle, measured in radians

Return value

If no errors occur, returns the value of the spherical associated Legendre function (that is, spherical harmonic with ϕ = 0) of l, m, and theta, where the spherical harmonic function is defined as Ym
l(theta,ϕ) = (-1)m
[

(2l+1)(l-m)!

4π(l+m)!

]1/2
Pm
l(cos(theta))eimϕ
where Pm
l(x) is std::assoc_legendre(l, m, x)) and |m|≤l.

Note that the Condon-Shortley phase term (-1)m
is included in this definition because it is omitted from the definition of Pm
l in std::assoc_legendre.

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 l≥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 the spherical harmonic function is available in boost.math, and it reduces to this function when called with the parameter phi set to zero.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::sph_legendre(int_num1, int_num2, num) has the same effect as std::sph_legendre(int_num1, int_num2, static_cast<double>(num)).

Example

Run this code

#include <cmath>
#include <iostream>
#include <numbers>
 
int main()
{
    // spot check for l=3, m=0
    double x = 1.2345;
    std::cout << "Y_3^0(" << x << ") = " << std::sph_legendre(3, 0, x) << '\n';
 
    // exact solution
    std::cout << "exact solution = "
              << 0.25 * std::sqrt(7 / std::numbers::pi)
                  * (5 * std::pow(std::cos(x), 3) - 3 * std::cos(x))
              << '\n';
}

Output:

Y_3^0(1.2345) = -0.302387
exact solution = -0.302387

See also

assoc_legendreassoc_legendrefassoc_legendrel (C++17)(C++17)(C++17)

associated Legendre polynomials (function)

External links

Weisstein, Eric W. "Spherical Harmonic." From MathWorld — A Wolfram Web Resource.