orthpoly::hermite -- the Hermite
polynomials
Introductionorthpoly::hermite(n,x) computes the value
of the n-th degree Hermite polynomial at the point
x.
Call(s)orthpoly::hermite(n, x)
Parametersn |
- | a nonnegative integer: the degree of the polynomial. |
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"). |
ReturnsIf 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 Hermite polynomial at this point is returned as an
arithmetical expression. If n is not a nonnegative
integer, then orthpoly::hermite returns itself
symbolically.
Details
Example
1Polynomials of domain type DOM_POLY are returned, if
identifiers or indexed identifiers are specified:
>> orthpoly::hermite(2, x)
2
poly(4 x - 2, [x])
>> orthpoly::hermite(3, x[1])
3
poly(8 x[1] - 12 x[1], [x[1]])
However, using arithmetical expressions as input the ``values'' of these polynomials are returned:
>> orthpoly::hermite(2, 6*x)
2
144 x - 2
>> orthpoly::hermite(3, x[1] + 2)
2 (x[1] + 2) (2 (x[1] + 2) (2 x[1] + 4) - 2) - 8 x[1] - 16
``Arithmetical expressions'' include numbers:
>> orthpoly::hermite(2, sqrt(2)), orthpoly::hermite(3, 8 + I), orthpoly::hermite(1000, 0.3);
6, 3808 + 1516 I, 2.26821486e1433
If no integer degree is specified, then
orthpoly::hermite returns itself symbolically:
>> orthpoly::hermite(n, x), orthpoly::hermite(1/2, x)
orthpoly::hermite(n, x), orthpoly::hermite(1/2, x)
BackgroundH(n,x)=2*x*H(n-1,x)-2*(n-1)*H(n-2,x)with H(0,x)=1 and H(1,x)=2*x.