std::laguerre, std::laguerref, std::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

 

Defined in header <cmath>
double      laguerre( unsigned int n, double x ); float       laguerre( unsigned int n, float x ); long 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)	(since C++17)
double      laguerre( unsigned int n, IntegralType x );	(2)	(since C++17)

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.

Parameters

n	-	the degree of the polymonial, 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 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 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

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_laguerreassoc_laguerrefassoc_laguerrel (C++17)(C++17)(C++17)

associated Laguerre polynomials (function)

External links

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