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.

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

Y m l (θ,ϕ) = (-1) m [(2l+1)(l-m)! 4π(l+m)!] 1/2 P m l (cosθ)e imϕ

where

P m l (x)

is std::assoc_legendre(l,m,x)) and

|m|\leql

Note that the Condon-Shortley phase term $(-1) m$ is included in this definition because it is omitted from the definition of $P m l$ 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 $l\geq128$ , 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

Run this code

#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

External links

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

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