Lazy Power Series Rings

• Introduction
• Creation of Lazy Series Rings
  • LazyPowerSeriesRing(C, n): Rng, RngIntElt → RngPowLaz
  • ChangeRing(L, C): RngPowLaz, Rng → RngPowLaz, Map
  • Example: Ring Create
• Functions on Lazy Series Rings
  • R . i: RngPowLaz, RngIntElt → RngPowLazElt
  • AssignNames(~R, S): RngPowLaz, [MonStgElt]
  • BaseRing(R): RngPowLaz → Rng
  • CoefficientRing(R): RngPowLaz → Rng
  • Rank(R): RngPowLaz → RngIntElt
  • R1 eq R2: RngPowLaz, RngPowLaz → BoolElt
  • Example: Ring Ops
• Elements
  • Creation of Finite Lazy Series
    • R ! c: RngPowLaz, RngElt → RngPowLazElt
    • R ! s: RngPowLaz, RngPowLazElt → RngPowLazElt
    • R ! S: RngPowLaz, [RngElt] → RngPowLazElt
    • LazySeries(R, f): RngPowLaz, RngMPolElt → RngPowLazElt
    • LazySeries(R, f): RngPowLaz, FldFunRatElt → RngPowLazElt
    • LazySeries(R, f): RngPowLaz, RngUPolElt → RngPowLazElt
    • Example: Elt Create Fin
    • Creation of Lazy Series from Maps
      • elt<R | m>: RngPowLaz, Map → RngPowLazElt
      • Example: Maps Uni
      • Example: Maps Multi
  • Arithmetic with Lazy Series
    • s + t: RngPowLazElt, RngPowLazElt → RngPowLazElt
    • - s: RngPowLazElt → RngPowLazElt
    • s - t: RngPowLazElt, RngPowLazElt → RngPowLazElt
    • s * t: RngPowLazElt, RngPowLazElt → RngPowLazElt
    • s + r: RngPowLazElt, RngElt → RngPowLazElt
    • r + s: RngElt, RngPowLazElt → RngPowLazElt
    • c * s: RngElt, RngPowLazElt → RngPowLazElt
    • s * c: RngPowLazElt, RngElt → RngPowLazElt
    • s * n: RngPowLazElt, [RngIntElt] → RngPowLazElt
    • s ^ n: RngPowLazElt, RngIntElt → RngPowLazElt
    • Example: Elt Arith
  • Finding Coefficients of Lazy Series
    • Coefficient(s, i): RngPowLazElt, RngIntElt → RngElt
    • Coefficient(s, T): RngPowLazElt, SeqEnum → RngElt
    • Coefficients(s, n): RngPowLazElt, RngIntElt → [RngElt]
    • Coefficients(s, l, n): RngPowLazElt, RngIntElt, RngIntElt → [RngElt]
    • Valuation(s): RngPowLazElt → Any
    • PrintToPrecision(s, n): RngPowLazElt, RngIntElt
    • PrintTermsOfDegree(s, l, n): RngPowLazElt, RngIntElt, RngIntElt
    • LeadingCoefficient(s): RngPowLazElt → RngElt
    • LeadingTerm(s): RngPowLazElt → RngPowLazElt
    • Example: Elt Coeffs Spiral
    • CoefficientsNonSpiral(s, n): RngPowLazElt, [RngIntElt] → SeqEnum
    • CoefficientsNonSpiral(s, n): RngPowLazElt, RngIntElt → SeqEnum
    • Index(s, i, n): RngPowLazElt, [RngIntElt], [RngIntElt] → RngIntElt
    • Example: Coeff Non Spiral
  • Predicates on Lazy Series
    • s eq t: RngPowLazElt, RngPowLazElt → BoolElt
    • IsZero(s): RngPowLazElt → BoolElt
    • IsOne(s): RngPowLazElt → BoolElt
    • IsMinusOne(s): RngPowLazElt → BoolElt
    • IsUnit(s): RngPowLazElt → BoolElt
    • IsWeaklyZero(s, n): RngPowLazElt, RngIntElt → BoolElt
    • IsWeaklyEqual(s, t, n): RngPowLazElt, RngPowLazElt, RngIntElt → BoolElt
  • Other Functions on Lazy Series
    • Derivative(s): RngPowLazElt → RngPowLazElt
    • Derivative(s, v): RngPowLazElt, RngIntElt → RngPowLazElt
    • Derivative(s, v, n): RngPowLazElt, RngIntElt, RngIntElt → RngPowLazElt
    • Integral(s): RngPowLazElt → RngPowLazElt
    • Integral(s, v): RngPowLazElt, RngIntElt → RngPowLazElt
    • Evaluate(s, t): RngPowLazElt, RngPowLazElt → RngPowLazElt
    • Evaluate(s, T): RngPowLazElt, [RngPowLazElt] → RngPowLazElt
    • Example: eval
    • SquareRoot(s): RngPowLazElt → RngPowLazElt
    • Sqrt(s): RngPowLazElt → RngPowLazElt
    • IsSquare(s): RngPowLazElt → BoolElt, RngPowLazElt
    • PolynomialCoefficient(s, i): RngPowLazElt, RngIntElt → RngPowLazElt
    • PolynomialCoefficient(s, i): RngPowLazElt, [RngIntElt] → RngPowLazElt