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

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Special mathematical 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)	(since C++17)
double assoc_laguerre( unsigned int n, unsigned int m, IntegralType x );	(2)	(since C++17)

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.

Parameters

n	-	the degree of the polymonial, 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 \((-1)^m \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{L}_{n+m}(x)\)(-1)m

dm

dxm

L
n+m(x), is returned (where \(\mathsf{L}_{n+m}(x)\)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 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 associated Laguerre polynomials are the polynomial solutions of the equation \(x\ddot{y} + (m+1-x)\dot{y} + ny = 0\)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

#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 (C++17)(C++17)(C++17)

Laguerre polynomials (function)

External links

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