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))));