weylalgebra1
http://www.math.kobe-u.ac.jp/OCD/weylalgebra1.tfb
2003-08-07
experimental
2002-08-07, 2003-11-28 revised to 1.1
1
1
freealg1
arith1
list1
relation1
This CD defines elements of the ring of differential operators
with coefficients in the polynomial ring.
diffop
constructor of a differential operator from a polynomial or
from an element of the finitely generated free algebra.
The inverse of gr.
d/dq q = q d/dq + 1
weylalgebra1.diffop( dq ~freealg1.times~ q ,
list1.list(list1.list(q),list1.list(dq)))
~relation1.eq~
weylalgebra1.diffop( (q ~arith1.times~ dq) ~arith1.plus~ 1,
list1.list(list1.list(q),list1.list(dq)));
gr
the symbol polynomial of a given differential operator.
The inverse of diffop.
$\gr( q \partial_{q} + 1) = q p + 1 $
weylalgebra1.gr( weylalgebra1.diffop( q ~arith1.times~ dq ~arith1.plus~ 1 ,
list1.list(list1.list(q),list1.list(dq))),
list1.list(list1.list(dq),list1.list(p)))
~relation1.eq~
(q ~arith1.times~ p ~arith1.plus~ 1);
diff
Differentiation of a given function in one variable.
$\frac{d x^2}{dx} = 2 x$
weylalgebra1.diff( x ~arith1.power~ 2 , x)
~relation1.eq~
( 2 ~arith1.times~ x );
partialdiff
partial differentiation of a given function.
$\frac{\partial^{2} x^{2} y}{\partial x^{2}} = 2 y $
weylalgebra1.partialdiff( x ~arith1.times~ x ~arith1.times~ y,
list1.list(list1.list(x,2)))
~relation1.eq~
( 2 ~arith1.times~ y );
times
multiplication in D
$\partial_{q} q = \partial{q} q + 1 $
dq ~weylalgebra1.times~ q
~relation1.eq~
(q ~weylalgebra1.times~ dq ~arith1.plus~ 1);
act
action of a differential operator to a function.
$ x^{m} \partial_{x}^{n} \partial_{y}^{r} \cdot f
= x^{m} \frac{partial^{n+r} f}{\partial x^{n} \partial y^{r}}
$
weylalgebra1.act(
weylalgebra1.diffop( (x ~arith1.power~ m) ~arith1.times~
(dx ~arith1.power~ n) ~arith1.times~
(dy ~arith1.power~ r),
list1.list(list1.list(x,y),list1.list(dx,dy))),
f) ~relation1.eq~
((x ~arith1.power~ m) ~arith1.times~
weylalgebra1.partialdiff(f, list1.list(list1.list(x,n),list1.list(y,r))));
act_of_poly
action of a polynomial as a differential operator to a function.
act_of_poly is equivalent to the composition of act and diffop.
$ x^{m} \partial_{x}^{n} \partial_{y}^{r} \cdot f
= x^{m} \frac{partial^{n+r} f}{\partial x^{n} \partial y^{r}}
$
weylalgebra.act_of_poly(
(x ~arith1.power~ m) ~arith1.times~
(dx ~arith1.power~ n) ~arith1.times~
(dy ~arith1.power~ r),
list1.list(list1.list(x,y),list1.list(dx,dy)),
f) ~relation1.eq~
((x ~arith1.power~ m) ~arith1.times~
weylalgebra.partialdiff(f, list1.list(list1.list(x,n),list1.list(y,r))));