Generalized Laguerre Function and Laguerre Polynomials
laguerreL(
returns
the Laguerre polynomial of degree n
,x
)n
if n
is
a nonnegative integer. When n
is not a nonnegative
integer, laguerreL
returns the Laguerre function.
For details, see Generalized Laguerre Function.
Find the Laguerre polynomial of degree 3
for
input 4.3
.
laguerreL(3,4.3)
ans = 2.5838
Find the Laguerre polynomial for symbolic inputs. Specify degree n
as 3
to
return the explicit form of the polynomial.
syms x laguerreL(3,x)
ans = - x^3/6 + (3*x^2)/2 - 3*x + 1
If the degree of the Laguerre polynomial n
is
not specified, laguerreL
cannot find the polynomial.
When laguerreL
cannot find the polynomial, it
returns the function call.
syms n x laguerreL(n,x)
ans = laguerreL(n, x)
Find the explicit form of the generalized Laguerre
polynomial L(n,a,x)
of degree n = 2
.
syms a x laguerreL(2,a,x)
ans = (3*a)/2 - x*(a + 2) + a^2/2 + x^2/2 + 1
When n
is not a nonnegative
integer, laguerreL(n,a,x)
returns the generalized
Laguerre function.
laguerreL(-2.7,3,2)
ans = 0.2488
laguerreL
is not defined for certain inputs
and returns an error.
syms x
laguerreL(-5/2, -3/2, x)
Error using symengine Function 'laguerreL' not supported for parameter values '-5/2' and '-3/2'.
Find the Laguerre polynomials of degrees 1
and 2
by
setting n = [1 2]
.
syms x laguerreL([1 2],x)
ans = [ 1 - x, x^2/2 - 2*x + 1]
laguerreL
acts element-wise on n
to
return a vector with two elements.
If multiple inputs are specified as a vector, matrix, or multidimensional
array, the inputs must be the same size. Find the generalized Laguerre
polynomials where input arguments n
and x
are
matrices.
syms a n = [2 3; 1 2]; xM = [x^2 11/7; -3.2 -x]; laguerreL(n,a,xM)
ans = [ a^2/2 - a*x^2 + (3*a)/2 + x^4/2 - 2*x^2 + 1,... a^3/6 + (3*a^2)/14 - (253*a)/294 - 676/1029] [ a + 21/5,... a^2/2 + a*x + (3*a)/2 + x^2/2 + 2*x + 1]
laguerreL
acts element-wise on n
and x
to
return a matrix of the same size as n
and x
.
Use limit
to find the
limit of a generalized Laguerre polynomial of degree 3
as x
tends
to ∞.
syms x expr = laguerreL(3,2,x); limit(expr,x,Inf)
ans = -Inf
Use diff
to find the third derivative of
the generalized Laguerre polynomial laguerreL(n,a,x)
.
syms n a expr = laguerreL(n,a,x); diff(expr,x,3)
ans = -laguerreL(n - 3, a + 3, x)
Use taylor
to find the
Taylor series expansion of the generalized Laguerre polynomial of
degree 2
at x = 0
.
syms a x expr = laguerreL(2,a,x); taylor(expr,x)
ans = (3*a)/2 - x*(a + 2) + a^2/2 + x^2/2 + 1
Plot the Laguerre polynomials of orders 1
through 4
.
syms x fplot(laguerreL(1:4,x)) axis([-2 10 -10 10]) grid on ylabel('L_n(x)') title('Laguerre polynomials of orders 1 through 4') legend('1','2','3','4','Location','best')
The generalized Laguerre function is not defined for
all values of parameters n
and a
because
certain restrictions on the parameters exist in the definition of
the hypergeometric functions. If the generalized Laguerre function
is not defined for a particular pair of n
and a
,
the laguerreL
function returns an error message.
See Return Generalized Laguerre Function.
The calls laguerreL(n,x)
and laguerreL(n,0,x)
are
equivalent.
If n
is a nonnegative integer,
the laguerreL
function returns the explicit form
of the corresponding Laguerre polynomial.
The special values are
implemented for arbitrary values of n
and a
.
If n
is a negative integer and a
is
a numerical noninteger value satisfying a ≥ -n,
then laguerreL
returns 0
.
If n
is a negative integer and a
is
an integer satisfying a <
-n, the function returns an
explicit expression defined by the reflection rule
If all arguments are numerical and at least one argument
is a floating-point number, then laguerreL(x)
returns
a floating-point number. For all other arguments, laguerreL(n,a,x)
returns
a symbolic function call.
chebyshevT
| chebyshevU
| gegenbauerC
| hermiteH
| hypergeom
| jacobiP
| legendreP