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

 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) (since C++17) double      assoc_legendre( unsigned int n, unsigned int m, Integral x ); (2) (since C++17)
1) Computes the associated Legendre polynomials of the degree n, order m, and argument x
4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

## Contents

### 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 Pm
n
of x, that is (1-x2
)m/2
 dm dxm
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:

• 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