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

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< cpp‎ | experimental‎ | 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.
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.

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 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)).

### 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 - 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

(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 legendref legendrel | Legendre polynomials (function)

### External links

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

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