# std::legendre, std::legendref, std::legendrel

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< cpp‎ | experimental‎ | special functions

double legendre( unsigned int n, double x ); double legendre( unsigned int n, float x ); double legendre( unsigned int n, long double x ); float legendref( unsigned int n, float x ); long double legendrel( unsigned int n, long double x ); | (1) |
double legendre( unsigned int n, IntegralType x ); | (2) |

1) Computes the unassociated Legendre polynomials of the degree n 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, `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
x | - | the argument, a value of a floating-point or integral type

### Return value

If no errors occur, value of the order-`n` unassociated Legendre polynomial of `x`, that is

1
2n n!

dn
dxn

(x2
- 1)n
, is returned.

### 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.
- The function is not required to be defined for |x| > 1.
- If n is greater or equal than 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 Legendre polynomials are:

- legendre(0, x) = 1.
- legendre(1, x) = x.
- legendre(2, x) =

  1
  2

  (3x2
  - 1).
- legendre(3, x) =

  1
  2

  (5x3
  - 3x).
- legendre(4, x) =

  1
  8

  (35x4
  - 30x2
  + 3).

### Example

(works as shown with gcc 6.0)

Run this code

    #define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
    #include <cmath>
    #include <iostream>

    double P3(double x)
    {
        return 0.5 * (5 * std::pow(x, 3) - 3 * x);
    }

    double P4(double x)
    {
        return 0.125 * (35 * std::pow(x, 4) - 30 * x * x + 3);
    }

    int main()
    {
        // spot-checks
        std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\n'
                  << std::legendre(4, 0.25) << '=' << P4(0.25) << '\n';
    }

Output:

    -0.335938=-0.335938
    0.157715=0.157715

### See also

laguerrelaguerreflaguerrel | Laguerre polynomials (function)
hermitehermitefhermitel | Hermite polynomials (function)

### External links

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

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