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Showing content from https://en.cppreference.com/w/cpp/algorithm/../numeric/special_functions/sph_legendre.html below:

std::sph_legendre, std::sph_legendref, std::sph_legendrel - cppreference.com

(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)

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.

[edit] Parameters l - degree m - order theta - polar angle, measured in radians [edit] Return value

If no errors occur, returns the value of the spherical associated Legendre function (that is, spherical harmonic with Ï = 0) of

l

m

, and

theta

, where the spherical harmonic function is defined as

Ym l(theta,Ï) = (-1)m []1/2 Pm l(cos(theta))eimÏ

where

Pm l(x)

std::assoc_legendre(l, m, x)

) and

|m|â¤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.

[edit] 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 lâ¥128, the behavior is implementation-defined.

[edit] 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)).

[edit] 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

[edit] See also [edit] External links

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