std::assoc_legendre, std::assoc_legendref, std::assoc_legendrel
double assoc_legendre( unsigned int n, unsigned int m, float x );
double assoc_legendre( unsigned int n, unsigned int m, long double x );
float assoc_legendref( unsigned int n, unsigned int m, float x );
As all special functions, assoc_legendre is only guaranteed to be available in <cmath> 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.
[edit] Parameters
[edit] Return value
If no errors occur, value of the associated Legendre polynomial Pmn of x, that is (1 - x2
)m/2
[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 |x| > 1, a domain error may occur.
- If
nis greater or equal to 128, the behavior is implementation-defined.
[edit] Notes
Implementations that do not support TR 29124 but support TR 19768, provide this function in the header tr1/cmath and namespace std::tr1.
An implementation of this function is also available in boost.math.
The first few associated Legendre polynomials are:
- assoc_legendre(0, 0, x) = 1.
- assoc_legendre(1, 0, x) = x.
- assoc_legendre(1, 1, x) = -(1 - x2
)1/2
. - assoc_legendre(2, 0, x) = 12(3x2
- 1). - assoc_legendre(2, 1, x) = -3x(1 - x2
)1/2
. - assoc_legendre(2, 2, x) = 3(1 - x2
).
[edit] Example
(works as shown with gcc 6.0)
#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1 #include <cmath> #include <iostream> double P20(double x) { return 0.5 * (3 * x * x - 1); } double P21(double x) { return -3.0 * x * std::sqrt (1 - x * x); } double P22(double x) { return 3 * (1 - x * x); } int main() { // spot-checks std::cout << std::assoc_legendre (2, 0, 0.5) << '=' << P20(0.5) << '\n' << std::assoc_legendre (2, 1, 0.5) << '=' << P21(0.5) << '\n' << std::assoc_legendre (2, 2, 0.5) << '=' << P22(0.5) << '\n'; }
Output:
-0.125=-0.125 -1.29904=-1.29904 2.25=2.25
[edit] See also
[edit] External links
Weisstein, Eric W. "Associated Legendre Polynomial." From MathWorld--A Wolfram Web Resource.