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
AsymptoticExpansionB()O()compare_with_values()error_part()exact_part()exp()factorial()has_same_summands()invert()is_exact()is_little_o_of_one()limit()log()map_coefficients()monomial_coefficient()plot_comparison()pow()rpow()show()sqrt()subs()summandssymbolic_expression()truncate()variable_names()
AsymptoticRingAsymptoticRingFunctorNoConvergenceError
- Common Asymptotic Expansions
- (Asymptotic) Growth Groups
- Description of Growth Groups
- Creating a Growth Group
- Classes and Methods
AbstractGrowthGroupFunctorDecreasingGrowthElementErrorExponentialGrowthElementExponentialGrowthGroupExponentialGrowthGroupFunctorExponentialNonGrowthElementExponentialNonGrowthGroupExponentialNonGrowthGroupFunctorGenericGrowthElementGenericGrowthGroupGenericNonGrowthElementGenericNonGrowthGroupGrowthGroupGrowthGroupFactorGrowthGroupFactoryMonomialGrowthElementMonomialGrowthGroupMonomialGrowthGroupFunctorMonomialNonGrowthElementMonomialNonGrowthGroupMonomialNonGrowthGroupFunctorPartialConversionElementPartialConversionValueErrorVariable
- Cartesian Products of Growth Groups
- (Asymptotic) Term Monoids
- Asymptotic Expansions — Miscellaneous
- Functions, Classes and Methods
LocalsNotImplementedBZeroNotImplementedOZeroWithLocalsbidirectional_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
FractionWithFactoredDenominatoralgebraic_dependence_certificate()algebraic_dependence_decomposition()asymptotic_decomposition()asymptotics()asymptotics_multiple()asymptotics_smooth()cohomology_decomposition()critical_cone()denominator()denominator_factored()denominator_ringdimension()grads()is_convenient_multiple_point()leinartas_decomposition()log_grads()maclaurin_coefficients()nullstellensatz_certificate()nullstellensatz_decomposition()numerator()numerator_ringquotient()relative_error()singular_ideal()smooth_critical_ideal()univariate_decomposition()
FractionWithFactoredDenominatorRingFractionWithFactoredDenominatorSumcoerce_point()diff_all()diff_op()diff_op_simple()diff_prod()diff_seq()direction()permutation_sign()subs_all()