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

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< cpp‎ | numeric‎ | special functions

Defined in header `<cmath>` |  |
| (1) |
float laguerre ( unsigned int n, float x ); double laguerre ( unsigned int n, double x ); long double laguerre ( unsigned int n, long double x ); |  | (since C++17) (until C++23)
/* floating-point-type */ laguerre( unsigned int n, /* floating-point-type */ x ); |  | (since C++23)
float laguerref( unsigned int n, float x ); | (2) | (since C++17)
long double laguerrel( unsigned int n, long double x ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Integer > double laguerre ( unsigned int n, Integer x ); | (A) | (since C++17)

1-3) Computes the non-associated Laguerre polynomials of the degree n and argument x. The library provides overloads of `std::laguerre` for all cv-unqualified floating-point types as the type of the parameter x. (since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

### Parameters

n | - | the degree of the polynomial, an unsigned integer value
x | - | the argument, a floating-point or integer value

### 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 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 Laguerre polynomials are the polynomial solutions of the equation .

The first few are:

Function | Polynomial
--- | ---
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)

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::laguerre(int_num, num) has the same effect as std::laguerre(int_num, static_cast<double>(num)).

### Example

Run this code

    #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'
                  << std::laguerre(3, 0.0) << '=' << 1.0 << '\n';
    }

Output:

    0.5=0.5
    0.125=0.125
    1=1

### See also

assoc_laguerre assoc_laguerref assoc_laguerrel (C++17)(C++17)(C++17) | associated Laguerre polynomials (function)

### External links

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

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