Asymptotic Expansions

The Asymptotic Ring

The asymptotic ring, as well as its main documentation is contained in the module

• Asymptotic Ring.

Asymptotic Expansion Generators

Some common asymptotic expansions can be generated in

• Common Asymptotic Expansions.

Supplements

Behind the scenes of working with asymptotic expressions a couple of additional classes and tools turn up. For instance the growth of each summand is managed in growth groups, see below.

Growth Groups

The growth of a summand of an asymptotic expression is managed in

• (Asymptotic) Growth Groups and

• Cartesian Products of Growth Groups.

Term Monoids

A summand of an asymptotic expression is basically a term out of the following monoid:

• (Asymptotic) Term Monoids.

Miscellaneous

Various useful functions and tools are collected in

• Asymptotic Expansions — Miscellaneous.

Asymptotic Expansions — Table of Contents

• Asymptotic Ring
  • (Informal) Definition
    • Growth Groups and Elements
  • Introductory Examples
    • Two Rings
      • A Univariate Asymptotic Ring
      • A Multivariate Asymptotic Ring
    • Arithmetical Operations
      • The Ring Operations Plus and Times
      • Division
      • Powers, Expontials and Logarithms
      • Multivariate Arithmetic
  • More Examples
    • The mathematical constant e as a limit
  • Selected Technical Details
    • Coercions and Functorial Constructions
    • Data Structures
  • Various
  • Classes and Methods
    • AsymptoticExpansion
      • AsymptoticExpansion.B()
      • AsymptoticExpansion.O()
      • AsymptoticExpansion.compare_with_values()
      • AsymptoticExpansion.error_part()
      • AsymptoticExpansion.exact_part()
      • AsymptoticExpansion.exp()
      • AsymptoticExpansion.factorial()
      • AsymptoticExpansion.has_same_summands()
      • AsymptoticExpansion.invert()
      • AsymptoticExpansion.is_exact()
      • AsymptoticExpansion.is_little_o_of_one()
      • AsymptoticExpansion.limit()
      • AsymptoticExpansion.log()
      • AsymptoticExpansion.map_coefficients()
      • AsymptoticExpansion.monomial_coefficient()
      • AsymptoticExpansion.plot_comparison()
      • AsymptoticExpansion.pow()
      • AsymptoticExpansion.rpow()
      • AsymptoticExpansion.show()
      • AsymptoticExpansion.sqrt()
      • AsymptoticExpansion.subs()
      • AsymptoticExpansion.substitute()
      • AsymptoticExpansion.summands
      • AsymptoticExpansion.symbolic_expression()
      • AsymptoticExpansion.truncate()
      • AsymptoticExpansion.variable_names()
    • AsymptoticRing
      • AsymptoticRing.B()
      • AsymptoticRing.Element
      • AsymptoticRing.change_parameter()
      • AsymptoticRing.coefficient_ring
      • AsymptoticRing.coefficients_of_generating_function()
      • AsymptoticRing.construction()
      • AsymptoticRing.create_summand()
      • AsymptoticRing.default_prec
      • AsymptoticRing.gen()
      • AsymptoticRing.gens()
      • AsymptoticRing.growth_group
      • AsymptoticRing.ngens()
      • AsymptoticRing.some_elements()
      • AsymptoticRing.term_monoid()
      • AsymptoticRing.term_monoid_factory
      • AsymptoticRing.variable_names()
    • AsymptoticRingFunctor
      • AsymptoticRingFunctor.merge()
    • NoConvergenceError
• Common Asymptotic Expansions
  • Classes and Methods
    • AsymptoticExpansionGenerators
      • AsymptoticExpansionGenerators.Binomial_kn_over_n()
      • AsymptoticExpansionGenerators.HarmonicNumber()
      • AsymptoticExpansionGenerators.ImplicitExpansion()
      • AsymptoticExpansionGenerators.ImplicitExpansionPeriodicPart()
      • AsymptoticExpansionGenerators.InverseFunctionAnalysis()
      • AsymptoticExpansionGenerators.SingularityAnalysis()
      • AsymptoticExpansionGenerators.Stirling()
      • AsymptoticExpansionGenerators.log_Stirling()
    • asymptotic_expansions
• (Asymptotic) Growth Groups
  • Description of Growth Groups
  • Creating a Growth Group
    • Some Examples
  • Classes and Methods
    • AbstractGrowthGroupFunctor
      • AbstractGrowthGroupFunctor.merge()
    • DecreasingGrowthElementError
    • ExponentialGrowthElement
      • ExponentialGrowthElement.base
    • ExponentialGrowthGroup
      • ExponentialGrowthGroup.DivisionRings
      • ExponentialGrowthGroup.Element
      • ExponentialGrowthGroup.Groups
      • ExponentialGrowthGroup.Magmas
      • ExponentialGrowthGroup.Posets
      • ExponentialGrowthGroup.Sets
      • ExponentialGrowthGroup.construction()
      • ExponentialGrowthGroup.factory()
      • ExponentialGrowthGroup.gens()
      • ExponentialGrowthGroup.non_growth_group()
      • ExponentialGrowthGroup.some_elements()
    • ExponentialGrowthGroupFunctor
    • ExponentialNonGrowthElement
    • ExponentialNonGrowthGroup
      • ExponentialNonGrowthGroup.Element
      • ExponentialNonGrowthGroup.construction()
    • ExponentialNonGrowthGroupFunctor
    • GenericGrowthElement
      • GenericGrowthElement.factors()
      • GenericGrowthElement.is_lt_one()
      • GenericGrowthElement.log()
      • GenericGrowthElement.log_factor()
      • GenericGrowthElement.rpow()
      • GenericGrowthElement.variable_names()
    • GenericGrowthGroup
      • GenericGrowthGroup.AdditiveMagmas
      • GenericGrowthGroup.Element
      • GenericGrowthGroup.Magmas
      • GenericGrowthGroup.Posets
      • GenericGrowthGroup.Sets
      • GenericGrowthGroup.extended_by_non_growth_group()
      • GenericGrowthGroup.gen()
      • GenericGrowthGroup.gens()
      • GenericGrowthGroup.gens_monomial()
      • GenericGrowthGroup.is_compatible()
      • GenericGrowthGroup.le()
      • GenericGrowthGroup.ngens()
      • GenericGrowthGroup.non_growth_group()
      • GenericGrowthGroup.some_elements()
      • GenericGrowthGroup.variable_names()
    • GenericNonGrowthElement
    • GenericNonGrowthGroup
    • GrowthGroup
    • GrowthGroupFactor
      • GrowthGroupFactor.base
      • GrowthGroupFactor.cls
      • GrowthGroupFactor.extend_by_non_growth_group
      • GrowthGroupFactor.var
    • GrowthGroupFactory
      • GrowthGroupFactory.create_key_and_extra_args()
      • GrowthGroupFactory.create_object()
    • MonomialGrowthElement
      • MonomialGrowthElement.exponent
    • MonomialGrowthGroup
      • MonomialGrowthGroup.AdditiveMagmas
      • MonomialGrowthGroup.Element
      • MonomialGrowthGroup.Magmas
      • MonomialGrowthGroup.Posets
      • MonomialGrowthGroup.Sets
      • MonomialGrowthGroup.construction()
      • MonomialGrowthGroup.factory()
      • MonomialGrowthGroup.gens_logarithmic()
      • MonomialGrowthGroup.gens_monomial()
      • MonomialGrowthGroup.non_growth_group()
    • MonomialGrowthGroupFunctor
    • MonomialNonGrowthElement
    • MonomialNonGrowthGroup
      • MonomialNonGrowthGroup.Element
      • MonomialNonGrowthGroup.construction()
    • MonomialNonGrowthGroupFunctor
    • PartialConversionElement
      • PartialConversionElement.is_compatible()
      • PartialConversionElement.split()
    • PartialConversionValueError
    • Variable
      • Variable.extract_variable_names()
      • Variable.is_monomial()
      • Variable.variable_names()
• Cartesian Products of Growth Groups
  • Classes and Methods
    • CartesianProductFactory
      • CartesianProductFactory.create_key_and_extra_args()
      • CartesianProductFactory.create_object()
    • GenericProduct
      • GenericProduct.Element
        • GenericProduct.Element.exp()
        • GenericProduct.Element.factors()
        • GenericProduct.Element.is_lt_one()
        • GenericProduct.Element.log()
        • GenericProduct.Element.log_factor()
        • GenericProduct.Element.rpow()
        • GenericProduct.Element.variable_names()
      • GenericProduct.cartesian_injection()
      • GenericProduct.gens_monomial()
      • GenericProduct.some_elements()
      • GenericProduct.variable_names()
    • MultivariateProduct
    • UnivariateProduct
• (Asymptotic) Term Monoids
  • Absorption of Asymptotic Terms
  • Comparison
  • Various
  • Classes and Methods
    • BTerm
      • BTerm.can_absorb()
      • BTerm.construction()
    • BTermMonoid
      • BTermMonoid.Element
      • BTermMonoid.some_elements()
    • DefaultTermMonoidFactory
    • ExactTerm
      • ExactTerm.can_absorb()
      • ExactTerm.is_constant()
      • ExactTerm.is_exact()
      • ExactTerm.is_little_o_of_one()
      • ExactTerm.log_term()
      • ExactTerm.rpow()
    • ExactTermMonoid
      • ExactTermMonoid.Element
    • GenericTerm
      • GenericTerm.absorb()
      • GenericTerm.can_absorb()
      • GenericTerm.construction()
      • GenericTerm.is_constant()
      • GenericTerm.is_exact()
      • GenericTerm.is_little_o_of_one()
      • GenericTerm.log_term()
      • GenericTerm.rpow()
      • GenericTerm.variable_names()
    • GenericTermMonoid
      • GenericTermMonoid.Element
      • GenericTermMonoid.change_parameter()
      • GenericTermMonoid.coefficient_ring
      • GenericTermMonoid.from_construction()
      • GenericTermMonoid.growth_group
      • GenericTermMonoid.le()
      • GenericTermMonoid.some_elements()
      • GenericTermMonoid.term_monoid()
      • GenericTermMonoid.term_monoid_factory
    • OTerm
      • OTerm.can_absorb()
      • OTerm.is_little_o_of_one()
      • OTerm.log_term()
      • OTerm.rpow()
    • OTermMonoid
      • OTermMonoid.Element
    • TermMonoidFactory
      • TermMonoidFactory.create_key_and_extra_args()
      • TermMonoidFactory.create_object()
    • TermWithCoefficient
      • TermWithCoefficient.construction()
    • TermWithCoefficientMonoid
      • TermWithCoefficientMonoid.Element
      • TermWithCoefficientMonoid.some_elements()
    • ZeroCoefficientError
    • absorption()
    • can_absorb()
• Asymptotic Expansions — Miscellaneous
  • Functions, Classes and Methods
    • Locals
      • Locals.default_locals()
    • NotImplementedBZero
    • NotImplementedOZero
    • WithLocals
      • WithLocals.locals()
    • bidirectional_merge_overlapping()
    • bidirectional_merge_sorted()
    • combine_exceptions()
    • log_string()
    • parent_to_repr_short()
    • repr_op()
    • repr_short_to_parent()
    • split_str_by_op()
    • strip_symbolic()
    • substitute_raise_exception()
    • transform_category()
• Asymptotics of Multivariate Generating Series
  • Introductory Examples
  • Various
  • Classes and Methods
    • FractionWithFactoredDenominator
      • FractionWithFactoredDenominator.algebraic_dependence_certificate()
      • FractionWithFactoredDenominator.algebraic_dependence_decomposition()
      • FractionWithFactoredDenominator.asymptotic_decomposition()
      • FractionWithFactoredDenominator.asymptotics()
      • FractionWithFactoredDenominator.asymptotics_multiple()
      • FractionWithFactoredDenominator.asymptotics_smooth()
      • FractionWithFactoredDenominator.cohomology_decomposition()
      • FractionWithFactoredDenominator.critical_cone()
      • FractionWithFactoredDenominator.denominator()
      • FractionWithFactoredDenominator.denominator_factored()
      • FractionWithFactoredDenominator.denominator_ring
      • FractionWithFactoredDenominator.dimension()
      • FractionWithFactoredDenominator.grads()
      • FractionWithFactoredDenominator.is_convenient_multiple_point()
      • FractionWithFactoredDenominator.leinartas_decomposition()
      • FractionWithFactoredDenominator.log_grads()
      • FractionWithFactoredDenominator.maclaurin_coefficients()
      • FractionWithFactoredDenominator.nullstellensatz_certificate()
      • FractionWithFactoredDenominator.nullstellensatz_decomposition()
      • FractionWithFactoredDenominator.numerator()
      • FractionWithFactoredDenominator.numerator_ring
      • FractionWithFactoredDenominator.quotient()
      • FractionWithFactoredDenominator.relative_error()
      • FractionWithFactoredDenominator.singular_ideal()
      • FractionWithFactoredDenominator.smooth_critical_ideal()
      • FractionWithFactoredDenominator.univariate_decomposition()
    • FractionWithFactoredDenominatorRing
      • FractionWithFactoredDenominatorRing.Element
      • FractionWithFactoredDenominatorRing.base_ring()
    • FractionWithFactoredDenominatorSum
      • FractionWithFactoredDenominatorSum.denominator_ring
      • FractionWithFactoredDenominatorSum.sum()
      • FractionWithFactoredDenominatorSum.whole_and_parts()
    • coerce_point()
    • diff_all()
    • diff_op()
    • diff_op_simple()
    • diff_prod()
    • diff_seq()
    • direction()
    • permutation_sign()
    • subs_all()

Indices and Tables

• Index

• Module Index

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