NAME¶

std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel - std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel

Synopsis¶

double assoc_laguerre ( unsigned int n, unsigned int m, double x );

double assoc_laguerre ( unsigned int n, unsigned int m, float x );
double assoc_laguerre ( unsigned int n, unsigned int m, long double x ); (1)
float assoc_laguerref( unsigned int n, unsigned int m, float x );

long double assoc_laguerrel( unsigned int n, unsigned int m, long double x );
double assoc_laguerre ( unsigned int n, unsigned int m, IntegralType x ); (2)

1) Computes the associated Laguerre polynomials of the degree n, order m, 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, assoc_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
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 Laguerre polynomial of x, that is (-1)m

dm
dxm

L
n + m(x), is returned (where L
n + m(x) is the unassociated Laguerre polynomial, std::laguerre(n + m, 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 is negative, a domain error may occur.
* If n or m 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 associated Laguerre polynomials are the polynomial solutions of the equation
xy,,
+ (m + 1 - x)y,
+ ny = 0.

The first few are:

* assoc_laguerre(0, m, x) = 1.
* assoc_laguerre(1, m, x) = -x + m + 1.
* assoc_laguerre(2, m, x) =

1
2

[x2
- 2(m + 2)x + (m + 1)(m + 2)].
* assoc_laguerre(3, m, x) =

1
6

[-x3
- 3(m + 3)x2
- 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3)].

Example¶

// Run this code

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

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

double L2(unsigned m, double x)
{
return 0.5 * (x * x - 2 * (m + 2) * x + (m + 1) * (m + 2));
}

int main()
{
// spot-checks
std::cout << std::assoc_laguerre(1, 10, 0.5) << '=' << L1(10, 0.5) << '\n'
<< std::assoc_laguerre(2, 10, 0.5) << '=' << L2(10, 0.5) << '\n';
}

Output:¶

10.5=10.5
60.125=60.125

See also¶

laguerre Laguerre polynomials
laguerref (function)
laguerrel

External links¶

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