std::assoc_laguerre,std::assoc_laguerref,std::assoc_laguerrel(3)

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

Synopsis
   Defined in header <cmath>
   double assoc_laguerre( unsigned int n, unsigned int m, double x );
   float assoc_laguerre( unsigned int n, unsigned int m, float x );
   long double assoc_laguerre( unsigned int n, unsigned int m, long
   double x ); (1) (since C++17)
   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, (2) (since C++17)
   IntegralType x );
   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
   laguerre
   laguerref
   laguerrel Laguerre polynomials
   (C++17) (function)
   (C++17)
   (C++17)

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