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

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Mathematical special functions
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)
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 Y ml (theta,ϕ) = (-1) m [
(2l+1)(l-m)!
4π(l+m)!
] 1/2 P ml (cos(theta))e imϕ where P ml (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 P ml 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

(C++17) (C++17) (C++17)
associated Legendre polynomials
(function)

External links

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