# Element Operations on Differential Operators ## Category and Parent ### `Category(L): RngDiffOpElt -> RngDiffOpElt` ### `Type(L): RngDiffOpElt -> RngDiffOpElt` The category, or type, of the differential operator $L$. ### `Parent(L): RngDiffOpElt -> RngDiffOp` The parent of the differential operator $L$. ## Arithmetic All the usual arithmetic operations are possible for differential operators. It follows from the multiplication rule for differential operators that the multiplication of differential operators is non–commutative. ### `s + t: RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt` ### `s + t: RngDiffOpElt, RngElt -> RngDiffOpElt` ### `s + t: RngElt, RngDiffOpElt -> RngDiffOpElt` The sum of the two differential operators $s$ and $t$. ### `- s: RngDiffOpElt -> RngDiffOpElt` The negation of the differential operator $s$. ### `s - t: RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt` ### `s - t: RngDiffOpElt, RngElt -> RngDiffOpElt` ### `s - t: RngElt, RngDiffOpElt -> RngDiffOpElt` The difference between the differential operators $s$ and $t$. ### `s * t: RngDiffOpElt, RngDiffOpElt -> RngDiffOpElt` ### `s * t: RngDiffOpElt, RngElt -> RngDiffOpElt` ### `s * t: RngElt, RngDiffOpElt -> RngDiffOpElt` The product of the differential operators $s$ and $t$. ### `s ^ n: RngDiffOpElt, RngIntElt -> RngDiffElt` Given a differential operator $s$ and an integer $n\ge 0$, return the $n$-th power of $s$. ### `Example: Diff Op Arithmetic (ex-a0ba79)` ```magma > F := RationalDifferentialField(Rationals()); > R := DifferentialOperatorRing(F); > (z*D-1)*(D+1); z*D^2 + (z - 1)*D + -1 > (D+1)*(z*D-1); z*D^2 + z*D + -1 > (D-1/z)^2; D^2 + -2/z*D + 2/z^2 ``` ## Predicates and Booleans ### `s eq t: RngDiffOpElt, RngDiffOpElt -> BoolElt` Return `true` iff the differential operators $s$ and $t$ are exactly the same. ### `IsZero(L): RngDiffOpElt -> BoolElt` Return `true` iff the differential operator $L$ is the zero element of its parent. ### `IsOne(L): RngDiffOpElt -> BoolElt` Return `true` iff the differential operator $L$ is the unity element of its parent. ### `IsMonic(L): RngDiffOpElt -> BoolElt` Return `true` iff the differential operator $L$ is monic. ### `IsWeaklyEqual(L, P): RngDiffOpElt, RngDiffOpElt -> BoolElt` Returns `true` if and only if the differential operator $L$ is weakly equal to the operator $P$. This means that the $i$-th coefficients of $L$ and $P$ should be weakly equal to each other for every $i \in [0..\max(\deg(L),\deg(P))]$. ### `IsWeaklyZero(L): RngDiffOpElt -> BoolElt` Returns `true` if and only if the differential operator $L\in R$ is weakly equal to $R ! 0$. ### `IsWeaklyMonic(L): RngDiffOpElt -> BoolElt` Returns `true` if and only if the leading coefficient of the differential operator $L$ is weakly equal to $1$. ## Coefficients and Terms Differential operators look like univariate polynomials with coefficients in a differential ring. Some of the terminology used for polynomial rings is mimicked for differential operators. ### `Eltseq(L): RngDiffOpElt -> SeqEnum` ### `Coefficients(L): RngDiffOpElt -> SeqEnum` Given an operator $L$ with coefficients in $R$, this function returns the sequence of elements in $R$, that are the coefficients of $L$. The sequence is ordered from the constant coefficient to the coefficient of the highest order term of $L$. ### `Coefficient(L, i): RngDiffOpElt, RngIntElt -> RngElt` Given an operator $L$ with coefficients in $R$, this function returns the coefficient of the monomial of degree $i$ in $L$, as an element of $R$. ### `LeadingCoefficient(L): RngDiffOpElt -> RngElt` Given an operator $L$ with coefficients in $R$, this function returns the coefficient of the highest order term of $L$. ### `LeadingTerm(L): RngDiffOpElt -> RngDiffOpElt` The leading term of the differential operator $L$. ### `Terms(L): RngDiffOpElt -> SeqEnum` Given an operator $L$ with coefficients in $R$, this function returns the sequence of non–zero coefficients of $L$ as elements of $R$. The sequence is ordered from the lowest order term to the highest order term in $L$. ### `Example: Diff Op Coeff Terms (ex-d50a45)` ```magma > F := RationalDifferentialField(Rationals()); > R := DifferentialOperatorRing(F); > L := D^3 + (-4*z + 5)*D + (3*z - 4); > L; D^3 + (-4*z + 5)*D + 3*z - 4 > Eltseq(L); [ 3*z - 4, -4*z + 5, 0, 1 ] > LeadingTerm(L); D^3 > Terms(L); [ 3*z - 4, (-4*z + 5)*D, D^3 ] ``` ## Order and Degree ### `Order(L): RngDiffOpElt -> RngIntElt` ### `Degree(L): RngDiffOpElt -> RngIntElt` Returns the order of the differential operator $L$. In the case that $L$ is identically $0$, the order is defined to be $-1$. ### `WeakOrder(L): RngDiffOpElt -> RngIntElt` ### `WeakDegree(L): RngDiffOpElt -> RngIntElt` If the differential operator $L$ is defined over a differential series ring, then the exponent of the highest coefficient of $L$ that is not weakly $0$ is returned. ### `Example Booleans Degrees Diff Ops (ex-873e6e)` ```magma > S := DifferentialLaurentSeriesRing(Rationals()); > R := DifferentialOperatorRing(S); > L := D^2 + 2*t; > P := O(t)*D^3 + (1+O(t))*D^2 + 2*t; > Order(L); 2 > Degree(P); 3 > L eq P; false > IsWeaklyEqual(L,P); true > WeakOrder(P); 2 ``` ## Related Differential Operators ### `MonicDifferentialOperator(L): RngDiffOpElt -> RngDiffOpElt` Given the differential operator $L$, this function returns the monic differential operator $1/c\cdot L$, where c is the leading coefficient of $L$. ### `Adjoint(L): RngDiffOpElt -> RngDiffOpElt` Returns the formal adjoint of the differential operator $L$. The *formal adjoint* of $L=\sum_{i=0}^n a_iD^i$ in the differential operator ring $R=F[D]$ over $F$, is the differential operator $L^*:=\sum_{i=0}^n (-1)^iD^i*a_i\in R$. It follows from the definition that the orders of $L$ and $L^*$ are the same and that the leading coefficient of $L^*$ is $(-1)^n a_n$. ### `Translation(L, e): RngDiffOpElt, RngElt -> RngDiffOpElt, Map` If $R$ is the parent of the differential operator $L$ and $e$ is a suitable ring element, then the operator in $R$ obtained by replacing $R.1$ by $R.1+e$ in $L$ is returned. The second argument returned is the translation map on $R$ by $e$. ### `TruncateCoefficients(L): RngDiffOpElt -> RngDiffOpElt` If $L$ is defined over a differential series ring, then returned is the operator whose coefficients are the truncations of the coefficients of $L$. ### `Example: Related Diff Op (ex-978632)` ```magma > F := RationalDifferentialField(Rationals()); > R := DifferentialOperatorRing(F); > L := z*D^3 + (-4*z + 5)*D + (3*z - 4); > Order(L); 3 > MonicDifferentialOperator(L); D^3 + (-4*z + 5)/z*D + (3*z - 4)/z > Adjoint(L); -z*D^3 + -3*D^2 + (4*z - 5)*D + 3*z > trans, mp := Translation(L, 2); > trans; z*D^3 + 6*z*D^2 + (8*z + 5)*D + 3*z + 6 ``` ### `Example: Related Diff Op Truncate Coefficients (ex-32cf07)` ```magma > S := DifferentialLaurentSeriesRing(Rationals()); > RS := DifferentialOperatorRing(S); > L := (5-O(t))*DS^3+(2*t^-1+t^2+O(t^4))*DS - t^-2+t+O(t^3); > L; (5 + O(t))*DS^3 + (2*t^-1 + t^2 + O(t^4))*DS + -t^-2 + t + O(t^3) > TruncateCoefficients(L); 5*DS^3 + (2*t^-1 + t^2)*DS + -t^-2 + t > L -TruncateCoefficients(L); O(t)*DS^3 + O(t^4)*DS + O(t^3) ``` ## Application of Operators As pointed out in the introduction a differential operator $L=a_nD^n+a_{n-1}D^{n-1}+\cdots+a_1D+a_0$ in $F[D]$ leads to the differential equation $L(y)=0$ given by $$ L(y)=a_n \delta_F^n(y)+a_{n-1} \delta_F^{n-1}(y)+ \cdots+a_1 \delta_F(y)+a_0 y $$ This notation is formal, but also defines an action of $L$ on any element $y\in F$. The function `Apply` returns the ring element obtained by this action. ### `Apply(L, f): RngDiffOpElt, RngElt -> RngElt` ### `L(f): RngElt, RngDiffOpElt -> RngElt` ### `f @ L: RngElt, RngDiffOpElt -> RngElt` Given a differential operator $L$ and a ring element $f$, return the ring element obtained after applying $L$ to $f$, as an element of the base ring of $L$. The element $f$ must be coercible into the base ring of $L$. ### `Example Apply (ex-1a4536)` ```magma > F := RationalDifferentialField(Rationals()); > R := DifferentialOperatorRing(F); > L := D^2-2/z^2; > Apply(L, z); -2/z > L(z); -2/z > Apply(L, z^2); 0 ```