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

 double      sph_legendre ( unsigned l, unsigned m, double θ ); double      sph_legendre ( unsigned l, unsigned m, float θ ); double      sph_legendre ( unsigned l, unsigned m, long double θ ); float       sph_legendref( unsigned l, unsigned m, float θ  ); long double sph_legendrel( unsigned l, unsigned m, long double θ ); (1) (since C++17) double      sph_legendre ( unsigned l, unsigned m, Integral θ ); (2) (since C++17)
1) Computes the spherical associated Legendre function of degree l, order m, and polar angle θ.
2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

Contents

Parameters

 l - degree m - order θ - 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 θ, where the spherical harmonic function is defined as Ym
l
(θ,ϕ) = (-1)m
[
 (2l+1)(l-m)! 4π(l+m)!
]1/2
Pm
l
(cosθ)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.

Example

#include <cmath>
#include <iostream>
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
double pi = std::acos(-1);
std::cout << "exact solution = "
<< 0.25*std::sqrt(7/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