The category, or type, of the differential ring element s.
The parent of the differential ring element s.
All the usual arithmetic operations are possible for differential ring elements.
The sum of the two differential ring elements s and t.
The negation of the differential ring element s.
The difference between the differential ring elements s and t.
The product of the differential ring elements s and t.
Given a differential ring element s and an integer n, return the n-th power of s. If s is invertible, n may be negative.
Given the differential ring elements s and t, return the exact division of s by t, if s is divisible by t.
Given the differential field elements s and t, return s divided by t.
Return true iff the differential ring elements s and t are exactly the same.
Return true iff the differential ring element s is the zero element of its parent.
Return true iff the differential ring element s is the unity element of its parent.
Return true if and only if the differential ring element s is weakly equal to the differential ring element t.
Return true if and only if the differential ring element s is weakly equal to the zero element of its parent.
Return true if and only if the differential ring element s is purely an order term of a differential series ring.
> F<z> := RationalDifferentialField(Rationals()); > S<t> := DifferentialLaurentSeriesRing(Rationals()); > IsOne(F!1); true > t eq t+O(t^2); false > IsWeaklyEqual(t, t+O(t^2)); true > IsWeaklyZero(t^(-1)); false > IsWeaklyZero(O(t)); true > IsOrderTerm(t+O(t^2)); false > IsOrderTerm(O(t)); true
Creates the order term of the differential series s.
The known part of the differential series s.
Returns the coefficients of the differential ring element s.
Returns the interval from the valuation of s to (including) the degree of s.
> F<z> := RationalDifferentialField(Rationals()); > _<X> := PolynomialRing(F); > K<x>, mp := ext<F|X^2+X+1>; > seq := Eltseq(x^2); > seq; [ -1, -1 ] > Universe(seq) eq F; true
> S<t> := DifferentialLaurentSeriesRing(Rationals()); > O(t+t^2); O(t) > Parent(O(t)) eq S; true > trunc := Truncate(t^(-1)+5*t^2 +O(t^4)); > trunc; t^-1 + 5*t^2 > Parent(trunc) eq S; true > seq := Eltseq(trunc); > seq; [ 1, 0, 0, 5 ] > Universe(seq) eq Rationals(); true > Exponents(trunc); [ -1 .. 2 ]
The minimal polynomial of the differential field element s over the base field.
> F<z> := RationalDifferentialField(Rationals()); > P<X> := PolynomialRing(F); > K<x>, mp := ext<F|X^2+X+1>; > f := MinimalPolynomial(x^2); > f; X^2 + X + 1 > Parent(f) eq P; true > g := MinimalPolynomial(x+3/2); > g; X^2 + -2*X + 7/4
The image of s under the derivation of the parent of s. Notice that it can be different to the "usual" derivative, as it relies on the defined derivation.
Returns the differential of s in the algebraic differential field F, as a differential in the differential space of the underlying ring of F.
> F<z> := RationalDifferentialField(Rationals()); > Derivative(z^2 + 7/z); (2*z^3 - 7)/z^2 > Differential(z); (1) d(z) > Differential(1/z+6+5*z); ((5*z^2 - 1)/z^2) d(z) > S<t> := DifferentialLaurentSeriesRing(Rationals()); > Derivative(5 + 2*t + 3*t^2); 2*t + 6*t^2