# Lazy Power Series Rings

- [Introduction](introduction.md)

- [Creation of Lazy Series Rings](creation.md)

  - [`LazyPowerSeriesRing(C, n): Rng, RngIntElt → RngPowLaz`](creation.md#function-lazypowerseriesring-rng-rngintelt)

  - [`ChangeRing(L, C): RngPowLaz, Rng → RngPowLaz, Map`](creation.md#function-changering-rngpowlaz-rng)

  - [`Example: Ring Create`](creation.md#example-ex-085add)

- [Functions on Lazy Series Rings](ring_ops.md)

  - [`R . i: RngPowLaz, RngIntElt → RngPowLazElt`](ring_ops.md#operation-operation-rngpowlaz-rngintelt-rngpowlazelt)

  - [`AssignNames(~R, S): RngPowLaz, [MonStgElt]`](ring_ops.md#function-assignnames-rngpowlaz-monstgelt-ref)

  - [`BaseRing(R): RngPowLaz → Rng`](ring_ops.md#function-basering-rngpowlaz)

  - [`CoefficientRing(R): RngPowLaz → Rng`](ring_ops.md#function-coefficientring-rngpowlaz)

  - [`Rank(R): RngPowLaz → RngIntElt`](ring_ops.md#function-rank-rngpowlaz)

  - [`R1 eq R2: RngPowLaz, RngPowLaz → BoolElt`](ring_ops.md#operation-op-eq-rngpowlaz-rngpowlaz)

  - [`Example: Ring Ops`](ring_ops.md#example-ex-aee740)

- [Elements](elements.md)

  - [Creation of Finite Lazy Series](elements.md#creation-of-finite-lazy-series)

    - [`R ! c: RngPowLaz, RngElt → RngPowLazElt`](elements.md#operation-op-rngpowlaz-rngelt)

    - [`R ! s: RngPowLaz, RngPowLazElt → RngPowLazElt`](elements.md#operation-op-rngpowlaz-rngpowlazelt)

    - [`R ! S: RngPowLaz, [RngElt] → RngPowLazElt`](elements.md#operation-op-rngpowlaz-rngelt-2)

    - [`LazySeries(R, f): RngPowLaz, RngMPolElt → RngPowLazElt`](elements.md#function-lazyseries-rngpowlaz-rngmpolelt)

    - [`LazySeries(R, f): RngPowLaz, FldFunRatElt → RngPowLazElt`](elements.md#function-lazyseries-rngpowlaz-fldfunratelt)

    - [`LazySeries(R, f): RngPowLaz, RngUPolElt → RngPowLazElt`](elements.md#function-lazyseries-rngpowlaz-rngupolelt)

    - [`Example: Elt Create Fin`](elements.md#example-ex-60e042)

    - [Creation of Lazy Series from Maps](elements.md#creation-of-lazy-series-from-maps)

      - [`elt<R | m>: RngPowLaz, Map → RngPowLazElt`](elements.md#constructor-constructor-elt-rngpowlaz-map-rngpowlazelt)

      - [`Example: Maps Uni`](elements.md#example-ex-6e6661)

      - [`Example: Maps Multi`](elements.md#example-ex-53315d)

  - [Arithmetic with Lazy Series](elements.md#arithmetic-with-lazy-series)

    - [`s + t: RngPowLazElt, RngPowLazElt → RngPowLazElt`](elements.md#operation-op-plus-rngpowlazelt-rngpowlazelt)

    - [`- s: RngPowLazElt → RngPowLazElt`](elements.md#operation-operation-rngpowlazelt-rngpowlazelt)

    - [`s - t: RngPowLazElt, RngPowLazElt → RngPowLazElt`](elements.md#operation-op-minus-rngpowlazelt-rngpowlazelt)

    - [`s * t: RngPowLazElt, RngPowLazElt → RngPowLazElt`](elements.md#operation-op-times-rngpowlazelt-rngpowlazelt)

    - [`s + r: RngPowLazElt, RngElt → RngPowLazElt`](elements.md#operation-op-plus-rngpowlazelt-rngelt)

    - [`r + s: RngElt, RngPowLazElt → RngPowLazElt`](elements.md#operation-op-plus-rngelt-rngpowlazelt)

    - [`c * s: RngElt, RngPowLazElt → RngPowLazElt`](elements.md#operation-op-times-rngelt-rngpowlazelt)

    - [`s * c: RngPowLazElt, RngElt → RngPowLazElt`](elements.md#operation-op-times-rngpowlazelt-rngelt)

    - [`s * n: RngPowLazElt, [RngIntElt] → RngPowLazElt`](elements.md#operation-op-times-rngpowlazelt-rngintelt)

    - [`s ^ n: RngPowLazElt, RngIntElt → RngPowLazElt`](elements.md#operation-op-pow-rngpowlazelt-rngintelt)

    - [`Example: Elt Arith`](elements.md#example-ex-75ff3a)

  - [Finding Coefficients of Lazy Series](elements.md#finding-coefficients-of-lazy-series)

    - [`Coefficient(s, i): RngPowLazElt, RngIntElt → RngElt`](elements.md#function-coefficient-rngpowlazelt-rngintelt)

    - [`Coefficient(s, T): RngPowLazElt, SeqEnum → RngElt`](elements.md#function-coefficient-rngpowlazelt-seqenum)

    - [`Coefficients(s, n): RngPowLazElt, RngIntElt → [RngElt]`](elements.md#function-rnglaz-coefficients)

    - [`Coefficients(s, l, n): RngPowLazElt, RngIntElt, RngIntElt → [RngElt]`](elements.md#function-coefficients-rngpowlazelt-rngintelt-rngintelt)

    - [`Valuation(s): RngPowLazElt → Any`](elements.md#function-valuation-rngpowlazelt)

    - [`PrintToPrecision(s, n): RngPowLazElt, RngIntElt`](elements.md#function-rnglaz-printtoprecision)

    - [`PrintTermsOfDegree(s, l, n): RngPowLazElt, RngIntElt, RngIntElt`](elements.md#function-printtermsofdegree-rngpowlazelt-rngintelt-rngintelt)

    - [`LeadingCoefficient(s): RngPowLazElt → RngElt`](elements.md#function-leadingcoefficient-rngpowlazelt)

    - [`LeadingTerm(s): RngPowLazElt → RngPowLazElt`](elements.md#function-leadingterm-rngpowlazelt)

    - [`Example: Elt Coeffs Spiral`](elements.md#example-ex-eaeab7)

    - [`CoefficientsNonSpiral(s, n): RngPowLazElt, [RngIntElt] → SeqEnum`](elements.md#function-rnglaz-cns)

    - [`CoefficientsNonSpiral(s, n): RngPowLazElt, RngIntElt → SeqEnum`](elements.md#function-coefficientsnonspiral-rngpowlazelt-rngintelt)

    - [`Index(s, i, n): RngPowLazElt, [RngIntElt], [RngIntElt] → RngIntElt`](elements.md#function-index-rngpowlazelt-rngintelt-rngintelt)

    - [`Example: Coeff Non Spiral`](elements.md#example-ex-31429a)

  - [Predicates on Lazy Series](elements.md#predicates-on-lazy-series)

    - [`s eq t: RngPowLazElt, RngPowLazElt → BoolElt`](elements.md#operation-op-eq-rngpowlazelt-rngpowlazelt)

    - [`IsZero(s): RngPowLazElt → BoolElt`](elements.md#function-iszero-rngpowlazelt)

    - [`IsOne(s): RngPowLazElt → BoolElt`](elements.md#function-isone-rngpowlazelt)

    - [`IsMinusOne(s): RngPowLazElt → BoolElt`](elements.md#function-isminusone-rngpowlazelt)

    - [`IsUnit(s): RngPowLazElt → BoolElt`](elements.md#function-isunit-rngpowlazelt)

    - [`IsWeaklyZero(s, n): RngPowLazElt, RngIntElt → BoolElt`](elements.md#function-isweaklyzero-rngpowlazelt-rngintelt)

    - [`IsWeaklyEqual(s, t, n): RngPowLazElt, RngPowLazElt, RngIntElt → BoolElt`](elements.md#function-isweaklyequal-rngpowlazelt-rngpowlazelt-rngintelt)

  - [Other Functions on Lazy Series](elements.md#other-functions-on-lazy-series)

    - [`Derivative(s): RngPowLazElt → RngPowLazElt`](elements.md#function-derivative-rngpowlazelt)

    - [`Derivative(s, v): RngPowLazElt, RngIntElt → RngPowLazElt`](elements.md#function-derivative-rngpowlazelt-rngintelt)

    - [`Derivative(s, v, n): RngPowLazElt, RngIntElt, RngIntElt → RngPowLazElt`](elements.md#function-derivative-rngpowlazelt-rngintelt-rngintelt)

    - [`Integral(s): RngPowLazElt → RngPowLazElt`](elements.md#function-integral-rngpowlazelt)

    - [`Integral(s, v): RngPowLazElt, RngIntElt → RngPowLazElt`](elements.md#function-integral-rngpowlazelt-rngintelt)

    - [`Evaluate(s, t): RngPowLazElt, RngPowLazElt → RngPowLazElt`](elements.md#function-evaluate-rngpowlazelt-rngpowlazelt)

    - [`Evaluate(s, T): RngPowLazElt, [RngPowLazElt] → RngPowLazElt`](elements.md#function-evaluate-rngpowlazelt-rngpowlazelt-2)

    - [`Example: eval`](elements.md#example-ex-f9b608)

    - [`SquareRoot(s): RngPowLazElt → RngPowLazElt`](elements.md#function-squareroot-rngpowlazelt)

    - [`Sqrt(s): RngPowLazElt → RngPowLazElt`](elements.md#function-sqrt-rngpowlazelt)

    - [`IsSquare(s): RngPowLazElt → BoolElt, RngPowLazElt`](elements.md#function-issquare-rngpowlazelt)

    - [`PolynomialCoefficient(s, i): RngPowLazElt, RngIntElt → RngPowLazElt`](elements.md#function-polynomialcoefficient-rngpowlazelt-rngintelt)

    - [`PolynomialCoefficient(s, i): RngPowLazElt, [RngIntElt] → RngPowLazElt`](elements.md#function-polynomialcoefficient-rngpowlazelt-rngintelt-2)