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

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Mathematical special functions
Defined in header <cmath>
(1)
float assoc_legendre ( unsigned int n, unsigned int m, float x ) ;

double assoc_legendre ( unsigned int n, unsigned int m, double x ) ;

long double assoc_legendre ( unsigned int n, unsigned int m, long double x ) ;
(since C++17)
(until C++23)
/* floating-point-type */ assoc_legendre ( unsigned int n, unsigned int m,
/* floating-point-type */ x ) ;
(since C++23)
float assoc_legendref ( unsigned int n, unsigned int m, float x ) ;
(2) (since C++17)
long double assoc_legendrel ( unsigned int n, unsigned int m, long double x ) ;
(3) (since C++17)
Defined in header <cmath>
template < class Integer >
double assoc_legendre ( unsigned int n, unsigned int m, Integer x ) ;
(A) (since C++17)
1-3) Computes the Associated Legendre polynomials of the degree n , order m , and argument x . The library provides overloads of std::assoc_legendre for all cv-unqualified floating-point types as the type of the parameter x . (since C++23)
A) Additional overloads are provided for all integer types, which are treated as double .

Parameters

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

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 ) ).

Note that the Condon-Shortley phase term (-1) m is omitted from this definition.

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 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 this function is also available in boost.math as boost::math::legendre_p , except that the boost.math definition includes the Condon-Shortley phase term.

The first few associated Legendre polynomials are:

Function Polynomial
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 )

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 :: assoc_legendre ( int_num1, int_num2, num ) has the same effect as std :: assoc_legendre ( int_num1, int_num2, static_cast < double > ( num ) ) .

Example

Run this code 
#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

(C++17) (C++17) (C++17)
Legendre polynomials
(function)

External links

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