std::sph_legendre,std::sph_legendref,std::sph_legendrel (3) - Linux Manuals

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

NAME

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

Synopsis

double sph_legendre ( unsigned l, unsigned m, double θ );
float sph_legendre ( unsigned l, unsigned m, float θ );
long double sph_legendre ( unsigned l, unsigned m, long double θ );(1) (since C++17)
float sph_legendref( unsigned l, unsigned m, float θ );
long double sph_legendrel( unsigned l, unsigned m, long double θ );
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

// Run this code

  #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.