W3cubDocs

/C++

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

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) 
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

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

© cppreference.com
Licensed under the Creative Commons Attribution-ShareAlike Unported License v3.0.
https://en.cppreference.com/w/cpp/numeric/special_functions/sph_legendre