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

 

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Mathematical special functions

assoc_laguerreassoc_laguerrefassoc_laguerrel assoc_legendreassoc_legendrefassoc_legendrel betabetafbetal comp_ellint_1comp_ellint_1fcomp_ellint_1l comp_ellint_2comp_ellint_2fcomp_ellint_2l comp_ellint_3comp_ellint_3fcomp_ellint_3l cyl_bessel_icyl_bessel_ifcyl_bessel_il

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl cyl_bessel_kcyl_bessel_kfcyl_bessel_kl cyl_neumanncyl_neumannfcyl_neumannl ellint_1ellint_1fellint_1l ellint_2ellint_2fellint_2l ellint_3ellint_3fellint_3l expintexpintfexpintl

hermitehermitefhermitel laguerrelaguerreflaguerrel legendrelegendreflegendrel riemann_zetariemann_zetafriemann_zetal sph_besselsph_besselfsph_bessell sph_legendresph_legendrefsph_legendrel sph_neumannsph_neumannfsph_neumannl

 

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

laguerrelaguerreflaguerrel

Laguerre polynomials (function)

External links

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