# diffOp(Ring,RingElement) -- create a differential operator from a Weyl algebra element

## Synopsis

• Function: diffOp
• Usage:
diffOp_R f
diffOp(f, R)
diffOp f
• Inputs:
• Outputs:
• Consequences:
• if called without a specified ring (i.e. diffOp f), creates a new ring by discarding the $dx$-variables and caches it in the Weyl algebra
• if such a ring had been previously cached, the cached ring will be used.

## Description

Creates a differential operator of the ring R from an element f of a Weyl algebra of R

 i1 : needsPackage "Dmodules" o1 = Dmodules o1 : Package i2 : R = QQ[x,y] o2 = R o2 : PolynomialRing i3 : S = makeWA R o3 = S o3 : PolynomialRing, 2 differential variables i4 : D = diffOp_R(x^2 * dx + y^2 * dy^2*dx) 2 2 2 o4 = y dx*dy + x dx o4 : DiffOp i5 : ring D === R o5 = true

The ring does not have to be specified. Note that in this case, the resulting operator will not be a differential operator of R, but that of a new ring. This ring is cached, so subsequent calls will result in operators of the same ring.

 i6 : E = diffOp(x^2* dx) 2 o6 = x dx o6 : DiffOp i7 : ring E === R o7 = false i8 : F = diffOp(dy^2) 2 o8 = dy o8 : DiffOp i9 : ring E === ring F o9 = true