orthpoly::laguerre
-- the
(generalized) Laguerre polynomialsorthpoly::laguerre
(n,a,x)
computes the
value of the generalized n-th degree Laguerre polynomial
with parameter a at the point x.
orthpoly::laguerre(n, a, x)
n |
- | a nonnegative integer: the degree of the polynomial. |
a |
- | an arithmetical expression. |
x |
- | an indeterminate or an arithmetical expression. An
indeterminate is either an identifier (of domain type DOM_IDENT ) or an indexed identifier (of type
"_index" ). |
If x
is an indeterminate, then a polynomial of domain
type DOM_POLY
is
returned. If x
is an arithmetical expression, then the
value of the Laguerre polynomial at this point is returned as an
arithmetical expression. If n
is not a nonnegative
integer, then orthpoly::laguerre
returns itself
symbolically.
Polynomials of domain type DOM_POLY
are returned, if
identifiers or indexed identifiers are specified:
>> orthpoly::laguerre(2, a, x)
/ / 2 \ \ | 2 | 3 a a | | poly| 1/2 x + (- a - 2) x + | --- + -- + 1 |, [x] | \ \ 2 2 / /
>> orthpoly::laguerre(3, a, x[1])
/ | 3 / a \ 2 poly| - 1/6 x[1] + | - + 3/2 | x[1] + \ \ 2 / / 2 \ / 3 \ \ | 5 a a | | 11 a 2 a | | | - --- - -- - 3 | x[1] + | ---- + a + -- + 1 |, [x[1]] | \ 2 2 / \ 6 6 / /
However, using arithmetical expressions as input the ``values'' of these polynomials are returned:
>> orthpoly::laguerre(2, 4, 6*x)
(5 - 6 x) (7 - 6 x) ------------------- - 5/2 2
>> orthpoly::laguerre(2, 2/3*I, x[1] + 2)
(- x[1] - (1 - 2/3 I)) ((1 + 2/3 I) - x[1]) ------------------------------------------- - (1/2 + 1/3 I) 2
``Arithmetical expressions'' include numbers:
>> orthpoly::laguerre(2, a, sqrt(2)), orthpoly::laguerre(3, 0.4, 8 + I), orthpoly::laguerre(1000, 3, 0.3);
2 3 a a 1/2 1/2 --- + -- - 2 2 - a 2 + 2, - 4.969333333 - 8.713333334 I, 2 2 -15691.69498
If no integer degree is specified, then
orthpoly::laguerre
returns itself symbolically:
>> orthpoly::laguerre(n, a, x), orthpoly::laguerre(1/2, a, x)
orthpoly::laguerre(n, a, x), orthpoly::laguerre(1/2, a, x)
L(n,a,x)=(2*n+a-1-x)/n*L(n-1,a,x)-(n+a-1)/n*L(n-2,a,x)with L(0,a,x)=1 and L(1,a,x)=1+a-x.