# std::laguerre, std::laguerref, std::laguerrel

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< cpp‎ | experimental‎ | special functions

double laguerre( unsigned int n, double x ); double laguerre( unsigned int n, float x ); double laguerre( unsigned int n, long double x ); float laguerref( unsigned int n, float x ); long double laguerrel( unsigned int n, long double x ); | (1) |
double laguerre( unsigned int n, IntegralType x ); | (2) |

1) Computes the non-associated Laguerre 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, `laguerre` 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
x | - | the argument, a value of a floating-point or integral type

### Return value

If no errors occur, value of the nonassociated Laguerre polynomial of `x`, that is

ex
n!

dn
dxn

(xn
e-x), 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.
- If x is negative, a domain error may occur.
- 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 Laguerre polynomials are the polynomial solutions of the equation xy,,
+ (1 - x)y,
+ ny = 0.

The first few are:

- laguerre(0, x) = 1.
- laguerre(1, x) = -x + 1.
- laguerre(2, x) =

  1
  2

  [x2
  - 4x + 2].
- laguerre(3, x) =

  1
  6

  [-x3
  - 9x2
  - 18x + 6].

### Example

(works as shown with gcc 6.0)

Run this code

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

    double L1(double x)
    {
        return -x + 1;
    }

    double L2(double x)
    {
        return 0.5 * (x * x - 4 * x + 2);
    }

    int main()
    {
        // spot-checks
        std::cout << std::laguerre(1, 0.5) << '=' << L1(0.5) << '\n'
                  << std::laguerre(2, 0.5) << '=' << L2(0.5) << '\n';
    }

Output:

    0.5=0.5
    0.125=0.125

### See also

assoc_laguerre assoc_laguerref assoc_laguerrel | associated Laguerre polynomials (function)

### External links

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

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