std::sph_legendre,std::sph_legendref,std::sph_legendrel(3)

NAME
std::sph_legendre,std::sph_legendref,std::sph_legendrel - std::sph_legendre,std::sph_legendref,std::sph_legendrel

Synopsis
   Defined in header <cmath>
   float sph_legendre ( unsigned l, unsigned m, float theta
   );
   double sph_legendre ( unsigned l, unsigned m, double (since C++17)
   theta ); (until C++23)
   long double sph_legendre ( unsigned l, unsigned m, long double
   theta );
   /* floating-point-type */ sph_legendre( unsigned l, unsigned
   m, (since C++23)
                        /* floating-point-type (1)
   */ theta );
   float sph_legendref( unsigned l, unsigned m, float theta (2) (since C++17)
   );
   long double sph_legendrel( unsigned l, unsigned m, long double (3) (since C++17)
   theta );
   Additional overloads
   Defined in header <cmath>
   template< class Integer >
   double sph_legendre ( unsigned l, unsigned m, Integer (A) (since C++17)
   theta );
   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 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
// 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
   assoc_legendre
   assoc_legendref
   assoc_legendrel associated Legendre polynomials
   (C++17) (function)
   (C++17)
   (C++17)

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