Asymptotic Expansions¶
The Asymptotic Ring¶
The asymptotic ring, as well as its main documentation is contained in the module
Asymptotic Expansion Generators¶
Some common asymptotic expansions can be generated in
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
Term Monoids¶
A summand of an asymptotic expression is basically a term out of the following monoid:
Miscellaneous¶
Various useful functions and tools are collected in
Asymptotic Expansions — Table of Contents¶
- Asymptotic Ring
- (Informal) Definition
- Introductory Examples
- More Examples
- Selected Technical Details
- Various
- Classes and Methods
AsymptoticExpansion
AsymptoticExpansion.B()
AsymptoticExpansion.O()
AsymptoticExpansion.compare_with_values()
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
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
- Classes and Methods
- (Asymptotic) Growth Groups
- Description of Growth Groups
- Creating a Growth Group
- Classes and Methods
AbstractGrowthGroupFunctor
DecreasingGrowthElementError
ExponentialGrowthElement
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
ExponentialNonGrowthGroupFunctor
GenericGrowthElement
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
GrowthGroupFactory
MonomialGrowthElement
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
MonomialNonGrowthGroupFunctor
PartialConversionElement
PartialConversionValueError
Variable
- Cartesian Products of Growth Groups
- (Asymptotic) Term Monoids
- Asymptotic Expansions — Miscellaneous
- Functions, Classes and Methods
- 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
FractionWithFactoredDenominatorSum
coerce_point()
diff_all()
diff_op()
diff_op_simple()
diff_prod()
diff_seq()
direction()
permutation_sign()
subs_all()