std:: assoc_legendre, std:: assoc_legendref, std:: assoc_legendrel

From cppreference.com
C++
Experimental
Mathematical special functions
double assoc_legendre ( unsigned int n, unsigned int m, double x ) ;

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 ) ;

long double assoc_legendrel ( unsigned int n, unsigned int m, long double x ) ;
(1)
double assoc_legendre ( unsigned int n, unsigned int m, IntegralType x ) ;
(2)
1) Computes the associated Legendre polynomials of the degree n , order m , and argument x .
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 .

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.

Parameters

n - the degree of the polynomial, a value of unsigned integer type
m - the order of the polynomial, a value of unsigned integer type
x - the argument, a value of a floating-point or integral type

Return value

If no errors occur, value of the associated Legendre polynomial P mn of x , that is (1 - x 2 ) m/2
d m
dx m
P n (x) , is returned (where P n (x) is the unassociated Legendre polynomial, std:: legendre ( n, x ) ).

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 n is greater or equal to 128, the behavior is implementation-defined.

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 - x 2 ) 1/2 .
  • assoc_legendre(2, 0, x) =
    1
    2
    (3x 2 - 1)     .
  • assoc_legendre(2, 1, x) = -3x(1 - x 2 ) 1/2 .
  • assoc_legendre(2, 2, x) = 3(1 - x 2 ) .

Example

(works as shown with gcc 6.0)

Run this code 
#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

See also

Legendre polynomials
(function)

External links

Weisstein, Eric W. "Associated Legendre Polynomial." From MathWorld--A Wolfram Web Resource.

Retrieved from " https://en.cppreference.com/mwiki/index.php?title=cpp/experimental/special_functions/assoc_legendre&oldid=169734 "