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

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< cpp‎ | numeric‎ | special math

double legendre( unsigned int n, double x ); float legendre( unsigned int n, float x ); long 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) | (since C++17)
double legendre( unsigned int n, IntegralType x ); | (2) | (since C++17)

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.

### 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 \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)

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 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

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

Run this code

    #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

laguerre laguerref laguerrel (C++17)(C++17)(C++17) | Laguerre polynomials (function)
hermite hermitef hermitel (C++17)(C++17)(C++17) | Hermite polynomials (function)

### External links

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

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