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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, IntegralType 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 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

External links

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