Laurent Series Ontology

flow "LaurentSeriesOntology" v1.0

# ============================================================
# LAURENT SERIES ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/complex-analysis/laurent-series-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Complex Analysis"
ENTITY := "Laurent Series"

# ---------- CORE ENTITY ----------
LAURENT_SERIES := {
    type: ComplexSeriesRepresentation,
    domain: ComplexAnalysis,
    purpose: "Represent a complex function on an annulus as a convergent series containing both nonnegative and negative integer powers about a centre."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    LAURENT_SERIES,
    COMPLEX_FUNCTION,
    ANNULUS,
    CENTRE,
    COEFFICIENT,
    PRINCIPAL_PART,
    ANALYTIC_PART,
    RESIDUE,
    ISOLATED_SINGULARITY,
    REMOVABLE_SINGULARITY,
    POLE,
    ESSENTIAL_SINGULARITY,
    CONTOUR_INTEGRAL,
    RADIUS_OF_CONVERGENCE,
    TAYLOR_SERIES
}

# ---------- BASIC FORM ----------
about centre
    z0

LAURENT_SERIES
    :=
sum_from_n_minus_infinity_to_infinity(
    a_n*(z-z0)^n
)

equivalently

f(z)
    =
sum_from_n_0_to_infinity(
    a_n*(z-z0)^n
)
+
sum_from_n_1_to_infinity(
    a_(-n)*(z-z0)^(-n)
)

# ---------- ANALYTIC PART ----------
ANALYTIC_PART
    :=
sum_from_n_0_to_infinity(
    a_n*(z-z0)^n
)

ANALYTIC_PART
    -> TAYLOR_LIKE_NONNEGATIVE_POWER_COMPONENT

# ---------- PRINCIPAL PART ----------
PRINCIPAL_PART
    :=
sum_from_n_1_to_infinity(
    a_(-n)*(z-z0)^(-n)
)

PRINCIPAL_PART
    -> encodes(SINGULAR_BEHAVIOUR_AT_CENTRE)

# ---------- ANNULUS OF CONVERGENCE ----------
LAURENT_SERIES
    -> converges_on(ANNULUS)

ANNULUS
    := {
        z :
        r < |z-z0| < R
    }

where

0 <= r < R <= infinity

INNER_RADIUS
    := r

OUTER_RADIUS
    := R

# ---------- UNIQUENESS ----------
if
    f holomorphic_on_annulus(
        r < |z-z0| < R
    )

then

f
    -> has(UNIQUE_LAURENT_SERIES_ABOUT_z0)
       on(that_annulus)

SAME_FUNCTION
    -> may_have(DIFFERENT_LAURENT_EXPANSIONS_ON_DIFFERENT_ANNULI)

# ---------- COEFFICIENT FORMULA ----------
for any positively_oriented_simple_closed_contour_C
inside annulus
winding_once_around(z0):

a_n
    =
1/(2*pi*i)
*
closed_integral_C(
    f(z)/(z-z0)^(n+1)
    dz
)

LAURENT_COEFFICIENT
    -> determined_by(CONTOUR_INTEGRAL)

# ---------- RESIDUE ----------
RESIDUE(f,z0)
    := a_(-1)

RESIDUE
    -> coefficient_of((z-z0)^(-1))

therefore

RESIDUE
    =
1/(2*pi*i)
*
closed_integral_C(
    f(z) dz
)

when
    contour_encloses_only_relevant_isolated_singularity_context

# ---------- TAYLOR SERIES RELATION ----------
if
    principal_part = 0
then

LAURENT_SERIES
    -> reduces_to(TAYLOR_SERIES)

TAYLOR_SERIES
    -> special_case_of(LAURENT_SERIES)

# ---------- ISOLATED SINGULARITY ----------
if
    f holomorphic_on(
        0 < |z-z0| < R
    )

then

z0
    -> ISOLATED_SINGULARITY_OR_REMOVABLE_POINT

LAURENT_SERIES
    -> classifies(BEHAVIOUR_AT_z0)

# ---------- REMOVABLE SINGULARITY ----------
if
    a_n = 0
    for_all n < 0
then

z0
    -> REMOVABLE_SINGULARITY

f
    -> extendable_holomorphically_to(z0)

PRINCIPAL_PART
    = 0

# ---------- POLE ----------
if
    principal_part_has_finitely_many_nonzero_terms

and

a_(-m) != 0

with
    a_n = 0
    for_all n < -m

then

z0
    -> POLE_OF_ORDER_m

POLE
    -> finite_principal_part

# ---------- SIMPLE POLE ----------
if
    only_negative_term_is

a_(-1)/(z-z0)

with
    a_(-1) != 0

then

z0
    -> SIMPLE_POLE

RESIDUE
    = a_(-1)

# ---------- ESSENTIAL SINGULARITY ----------
if
    principal_part_has_infinitely_many_nonzero_terms
then

z0
    -> ESSENTIAL_SINGULARITY

ESSENTIAL_SINGULARITY
    -> infinite_negative_power_structure

# ---------- ZERO / POLE FACTORISATION ----------
if
    f has pole_of_order_m at z0

then

f(z)
    =
(z-z0)^(-m) * g(z)

where

g holomorphic_near(z0)
AND
g(z0) != 0

LAURENT_SERIES
    -> exposes(POLE_ORDER)

# ---------- DERIVATION FROM CAUCHY INTEGRAL ----------
for
    f holomorphic_on_annulus

CAUCHY_KERNEL
    :=
1/(w-z)

depending_on(relative_radii)

-> expand_as(GEOMETRIC_SERIES)

GEOMETRIC_EXPANSION
    -> produces(POSITIVE_OR_NEGATIVE_POWERS)

CAUCHY_INTEGRAL_METHOD
    -> derives(LAURENT_EXPANSION)

# ---------- INNER / OUTER EXPANSIONS ----------
for rational expression:

choice_of_expansion
    -> depends_on(|z-z0| relative_to singularity_distance)

example_structure:

1/(z-a)

may_expand_in
    positive_powers_of(z-z0)
    or
    negative_powers_of(z-z0)

depending_on(ANNULUS)

ANNULUS
    -> essential_to(CORRECT_LAURENT_FORM)

# ---------- DIFFERENTIATION ----------
inside annulus of convergence:

d/dz[
    sum_n(
        a_n*(z-z0)^n
    )
]
=
sum_n(
    n*a_n*(z-z0)^(n-1)
)

LAURENT_SERIES
    -> differentiable_term_by_term

DERIVATIVE_SERIES
    -> converges_on(SAME_ANNULUS)

# ---------- INTEGRATION ----------
termwise integration:

for
    n != -1

integral(
    a_n*(z-z0)^n dz
)
=
a_n/(n+1)
*
(z-z0)^(n+1)

for
    n = -1

integral(
    a_(-1)/(z-z0) dz
)
=
a_(-1)*Log(z-z0)

LOG_TERM
    -> signals(NONZERO_RESIDUE_AND_MULTIVALUED_PRIMITIVE)

# ---------- PRIMITIVE CONDITION ----------
holomorphic function on annulus
    -> has(single_valued_primitive_on_annulus)

only_if

RESIDUE_AROUND_HOLE
    = 0
       for_relevant_cycle

NONZERO_a_minus_1
    -> obstruction_to(GLOBAL_PRIMITIVE_ON_PUNCTURED_REGION)

# ---------- CONTOUR INTEGRAL ----------
for circle
    |z-z0|=rho
    within annulus

closed_integral(
    (z-z0)^n dz
)

=
0
    if n != -1

=
2*pi*i
    if n = -1

therefore

CONTOUR_INTEGRAL
    -> extracts(a_minus_1)

# ---------- RESIDUE THEOREM RELATION ----------
LAURENT_SERIES
    -> provides(LOCAL_RESIDUES)

RESIDUE_THEOREM
    -> sums(LOCAL_a_minus_1_COEFFICIENTS)

LOCAL_SERIES_DATA
    -> determines(GLOBAL_CONTOUR_INTEGRAL)

# ---------- CONVERGENCE ----------
LAURENT_SERIES
    -> converges_absolutely
       on(interior_points_of_annulus)

LAURENT_SERIES
    -> converges_uniformly
       on(COMPACT_SUBANNULI)

UNIFORM_CONVERGENCE
    -> supports(TERM_BY_TERM_OPERATIONS)

# ---------- APPLICATIONS ----------
LAURENT_SERIES
    -> applied_to(SINGULARITY_CLASSIFICATION)
    -> applied_to(RESIDUE_COMPUTATION)
    -> applied_to(CONTOUR_INTEGRATION)
    -> applied_to(ANALYTIC_CONTINUATION)
    -> applied_to(MEROMORPHIC_FUNCTIONS)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
COMPLEX_FUNCTION
    + ANNULUS_OF_HOLOMORPHY
    -> LAURENT_SERIES

LAURENT_SERIES
    -> decomposes_into(ANALYTIC_PART, PRINCIPAL_PART)

PRINCIPAL_PART
    -> classifies(SINGULARITY)

a_minus_1
    -> RESIDUE

RESIDUE
    -> contributes_to(CONTOUR_INTEGRAL)

ZERO_PRINCIPAL_PART
    -> REMOVABLE_SINGULARITY

FINITE_PRINCIPAL_PART
    -> POLE

INFINITE_PRINCIPAL_PART
    -> ESSENTIAL_SINGULARITY

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Laurent series" ~= "two-sided complex power series with positive and negative integer powers",
    "principal part" ~= "negative-power portion of a Laurent expansion",
    "analytic part" ~= "nonnegative-power portion of a Laurent expansion",
    "Laurent coefficient" ~= "coefficient obtained by a contour integral around the expansion centre",
    "residue" ~= "coefficient a_-1 of the Laurent series",
    "annulus of convergence" ~= "ring-shaped domain on which a Laurent expansion converges"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    laurent_expansion_is_annulus_specific: true,
    same_function_may_have_different_expansions_on_different_annuli: true,
    residue_is_exactly_negative_first_power_coefficient: true,
    removable_singularity_has_zero_principal_part: true,
    pole_has_finite_nonzero_principal_part: true,
    essential_singularity_has_infinite_principal_part: true,
    termwise_operations_require_points_within_annulus_of_convergence: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    annulus_relation: 0.99,
    coefficient_formula: 0.99,
    singularity_classification: 0.99,
    residue_relation: 0.99,
    convergence_operation_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_and_expansion_centre)
    -> locate(singularities_relative_to_centre)
    -> determine(target_annulus)
    -> choose(geometric_or_contour_expansion_method)
    -> construct(positive_and_negative_power_terms)
    -> identify(principal_and_analytic_parts)
    -> extract(residue_coefficient)
    -> classify(central_singularity_if_applicable)
    -> validate(annulus_of_convergence)
    -> apply(termwise_differentiation_or_integration_if_requested)
    -> connect_to(residue_theorem_if_contour_problem)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Laurent-series semantic graph",
    primary_entity: LAURENT_SERIES,
    preserve: [
        function,
        expansion_centre,
        inner_radius,
        outer_radius,
        coefficients,
        analytic_part,
        principal_part,
        residue
    ],
    infer_through: [
        annuli,
        geometric_series,
        contour_coefficient_formulas,
        removable_singularities,
        poles,
        essential_singularities,
        residues,
        contour_integrals
    ],
    avoid: [
        giving_laurent_series_without specifying annulus,
        using_expansion_valid_for wrong radial region,
        confusing_taylor_and_laurent series,
        misidentifying_residue_as constant term,
        classifying_pole_without finite principal part,
        performing_termwise_operations_outside convergence annulus
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(laurent_series_context)
    -> compare(existing_relations)
    -> preserve(centre_annulus_coefficient_and_principal_part_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + LAURENT_SERIES
    + ANNULUS
    + ANALYTIC_PART
    + PRINCIPAL_PART
    + RESIDUE
    + REMOVABLE_SINGULARITY
    + POLE
    + ESSENTIAL_SINGULARITY
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> LAURENT_SERIES
    -> FUNCTION_CENTRE_IDENTIFICATION
    -> SINGULARITY_LOCATION_ANALYSIS
    -> ANNULUS_SELECTION
    -> EXPANSION_METHOD_SELECTION
    -> SERIES_CONSTRUCTION
    -> PRINCIPAL_ANALYTIC_PART_ANALYSIS
    -> RESIDUE_EXTRACTION
    -> SINGULARITY_CLASSIFICATION
    -> CONVERGENCE_VALIDATION
    -> TERM_OPERATION_ANALYSIS
    -> RESIDUE_THEOREM_RELATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW