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

double      assoc_legendre( unsigned int n, unsigned int m, double x );
float       assoc_legendre( unsigned int n, unsigned int m, float x );
long 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) (since C++17) 
double      assoc_legendre( unsigned int n, unsigned int m, IntegralType x );
 (2) (since C++17)
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.

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 \(\mathsf{P}_n^m\)Pm
n of x, that is \((1 - x^2) ^ {m/2} \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{P}_n(x)\)(1-x2
)m/2
dm
dxm
P
n(x), is returned (where \(\mathsf{P}_n(x)\)P
n(x) is the unassociated Legendre polynomial, std::legendre(n, x)).

Note that the Condon-Shortley phase term \((-1)^m\)(-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:

  • 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) =
    1
    2
    (3x2
    -1)
  • assoc_legendre(2, 1, x) = 3x(1-x2
    )1/2
  • assoc_legendre(2, 2, x) = 3(1-x2
    )

Example

#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)

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

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