std::sph_legendre, std::sph_legendref, std::sph_legendrel
Defined in header <cmath>
|
||
double sph_legendre ( unsigned l, unsigned m, double θ ); float sph_legendre ( unsigned l, unsigned m, float θ ); |
(1) | (since C++17) |
double sph_legendre ( unsigned l, unsigned m, IntegralType θ ); |
(2) | (since C++17) |
Contents |
[edit] Parameters
l | - | degree |
m | - | order |
θ | - | 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 ϕ = 0) ofl
, m
, and θ
, where the spherical harmonic function is defined as Yml(θ,ϕ) = (-1)m
[
(2l+1)(l-m)! |
4π(l+m)! |
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.
[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 l≥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.
[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
(C++17)(C++17)(C++17) |
associated Legendre polynomials (function) |
[edit] External links
Weisstein, Eric W. "Spherical Harmonic." From MathWorld — A Wolfram Web Resource. |