Analytic Function Ontology

flow "AnalyticFunctionOntology" v1.0

# ============================================================
# ANALYTIC FUNCTION ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/complex-analysis/analytic-function-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Complex Analysis"
ENTITY := "Analytic Function"

# ---------- CORE ENTITY ----------
ANALYTIC_FUNCTION := {
    type: ComplexFunctionProperty,
    domain: ComplexAnalysis,
    purpose: "Represent a function that is locally expressible as a convergent power series, equivalent in complex analysis to holomorphicity on an open domain."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    ANALYTIC_FUNCTION,
    HOLOMORPHIC_FUNCTION,
    COMPLEX_FUNCTION,
    POWER_SERIES,
    TAYLOR_SERIES,
    COMPLEX_DERIVATIVE,
    RADIUS_OF_CONVERGENCE,
    ANALYTIC_CONTINUATION,
    IDENTITY_THEOREM,
    ZERO,
    SINGULARITY,
    CAUCHY_INTEGRAL_FORMULA,
    ENTIRE_FUNCTION
}

# ---------- BASIC DEFINITION ----------
f
    -> ANALYTIC_AT(z0)

iff

exists
    r > 0

and coefficients
    a_n

such_that

for
    |z-z0| < r

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

ANALYTICITY
    -> LOCAL_POWER_SERIES_REPRESENTABILITY

# ---------- ANALYTIC ON OPEN SET ----------
f
    -> ANALYTIC_ON(D)

if

for_every
    z0 in D

f
    -> analytic_at(z0)

D
    -> OPEN_DOMAIN
       in(STANDARD_COMPLEX_ANALYSIS_CONTEXT)

# ---------- HOLOMORPHIC EQUIVALENCE ----------
in complex analysis:

f holomorphic_on(D)

iff

f analytic_on(D)

HOLOMORPHICITY
    <-> COMPLEX_ANALYTICITY

COMPLEX_DIFFERENTIABILITY_ON_OPEN_SET
    -> POWER_SERIES_STRUCTURE

# ---------- TAYLOR COEFFICIENTS ----------
if
    f analytic_at(z0)

then

a_n
    =
f^(n)(z0)/n!

therefore

f(z)
    =
sum_from_n_0_to_infinity(
    f^(n)(z0)/n!
    * (z-z0)^n
)

TAYLOR_SERIES
    -> uniquely_determined_by(DERIVATIVES_AT_CENTRE)

# ---------- INFINITE DIFFERENTIABILITY ----------
ANALYTIC_FUNCTION
    -> COMPLEX_DIFFERENTIABLE_OF_ALL_ORDERS

f^(n)
    -> ANALYTIC

DERIVATIVES
    -> preserve(ANALYTICITY)

# ---------- CAUCHY INTEGRAL FORMULA ----------
if
    f holomorphic
    on neighbourhood_of_closed_contour_region

then

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

CAUCHY_FORMULA
    -> generates(ALL_DERIVATIVES)

CAUCHY_FORMULA
    -> establishes(HOLOMORPHIC_IMPLIES_ANALYTIC)

# ---------- RADIUS OF CONVERGENCE ----------
TAYLOR_SERIES_AT(z0)
    -> has(RADIUS_OF_CONVERGENCE_R)

R
    -> constrained_by(distance_to_boundary_or_nearest_singularity)
       subject_to(NEAREST_SINGULARITY_STRUCTURE)

more precisely:

R
    -> distance_from(z0)
       to(nearest_obstruction_to_analytic_continuation)
       in(appropriate_complex_plane_context)

# ---------- POWER SERIES PROPERTIES ----------
inside radius R:

POWER_SERIES
    -> converges_absolutely
    -> converges_uniformly_on(COMPACT_SUBDISKS)
    -> may_be_differentiated_TERM_BY_TERM
    -> may_be_integrated_TERM_BY_TERM

DERIVED_SERIES
    -> same_radius_of_convergence

# ---------- LOCALITY ----------
ANALYTICITY
    -> LOCAL_PROPERTY

if
    every_point_has_neighbourhood_where_f_is_analytic
then

f
    -> ANALYTIC_ON_DOMAIN

LOCAL_POWER_SERIES
    -> patch_together_by(UNIQUENESS)

# ---------- IDENTITY THEOREM ----------
if
    f and g
        -> analytic_on(CONNECTED_DOMAIN_D)

and

set
    {
        z :
        f(z)=g(z)
    }

has
    accumulation_point_in(D)

then

f
    = g
    on(D)

ANALYTIC_FUNCTIONS
    -> RIGIDLY_DETERMINED_BY_LOCAL_DATA

# ---------- ZERO SET ----------
if
    f analytic_on(connected_domain)
    AND
    f not_identically_zero
then

ZEROS(f)
    -> ISOLATED

ACCUMULATION_OF_ZEROS_INSIDE_DOMAIN
    -> implies(f_identically_zero)

# ---------- ZERO MULTIPLICITY ----------
if
    f(z0)=0
    AND f not_identically_zero
then

exists
    m >= 1

such_that

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

where
    g analytic_near(z0)
    AND g(z0) != 0

m
    -> ORDER_OF_ZERO

# ---------- ANALYTIC CONTINUATION ----------
if
    f analytic_on(D1)

and
    g analytic_on(D2)

and
    f=g
    on(nonempty_open_overlap)

then

g
    -> ANALYTIC_CONTINUATION_OF(f)
       onto(D2)

IDENTITY_THEOREM
    -> guarantees(UNIQUENESS_ON_CONNECTED_CONTINUATION)

# ---------- MAXIMAL ANALYTIC CONTINUATION ----------
ANALYTIC_GERM
    -> may_be_extended_along(PATHS)

MAXIMAL_CONTINUATION
    -> limited_by(SINGULARITIES_OR_TOPOLOGICAL_OBSTRUCTIONS)

ANALYTIC_CONTINUATION
    -> may_be(MULTIVALUED_WITH_MONODROMY)

# ---------- SINGULARITY ----------
z0
    -> SINGULARITY

if
    analytic_continuation_fails_at(z0)

ISOLATED_SINGULARITY
    -> may_be(REMOVABLE)
    -> may_be(POLE)
    -> may_be(ESSENTIAL)

SINGULARITIES
    -> limit(TAYLOR_RADIUS)

# ---------- ENTIRE FUNCTION ----------
f
    -> ENTIRE

iff

f analytic_on(C)

ENTIRE_FUNCTION
    -> Taylor_series_about_any_point_has(INFINITE_RADIUS)
       when(no_finite_complex_singularities)

examples:
    exp(z)
    sin(z)
    cos(z)
    polynomials

# ---------- ANALYTIC ALGEBRA ----------
if
    f,g analytic_on(D)

then

f+g
    -> analytic

f-g
    -> analytic

f*g
    -> analytic

f/g
    -> analytic_where(g != 0)

ANALYTIC_FUNCTIONS
    -> closed_under(ALGEBRAIC_OPERATIONS_WHERE_DEFINED)

# ---------- COMPOSITION ----------
if
    f analytic_on(D)
    AND
    g analytic_on(E)
    AND
    f(D) subset_of E
then

g composed_with f
    -> ANALYTIC_ON(D)

CHAIN_RULE
    -> preserves(ANALYTICITY_UNDER_COMPOSITION)

# ---------- LOCAL INVERSE ----------
if
    f analytic_near(z0)
    AND
    f_prime(z0) != 0
then

f
    -> LOCALLY_BIHOLOMORPHIC

inverse
    -> ANALYTIC

ANALYTIC_INVERSE_FUNCTION_THEOREM
    -> local_inverse_structure

# ---------- REAL ANALYTIC DISTINCTION ----------
REAL_ANALYTIC_FUNCTION
    -> locally_representable_by(CONVERGENT_REAL_POWER_SERIES)

SMOOTH_REAL_FUNCTION
    -/> necessarily(REAL_ANALYTIC)

in complex analysis:

HOLOMORPHIC
    -> ANALYTIC

COMPLEX_ANALYTICITY
    -> much_more_rigid_than(C_INFINITY_REAL_SMOOTHNESS)

# ---------- APPLICATIONS ----------
ANALYTIC_FUNCTION
    -> applied_to(COMPLEX_ANALYSIS)
    -> applied_to(CONTOUR_INTEGRATION)
    -> applied_to(ANALYTIC_CONTINUATION)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(SPECIAL_FUNCTIONS)
    -> applied_to(NUMBER_THEORY)
    -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
COMPLEX_DIFFERENTIABILITY_ON_OPEN_SET
    -> HOLOMORPHIC_FUNCTION

HOLOMORPHIC_FUNCTION
    -> ANALYTIC_FUNCTION

ANALYTIC_FUNCTION
    -> has(LOCAL_TAYLOR_SERIES)

TAYLOR_COEFFICIENTS
    -> determined_by(DERIVATIVES)

SINGULARITY
    -> limits(RADIUS_OF_CONVERGENCE)

IDENTITY_THEOREM
    -> ensures(UNIQUENESS_OF_ANALYTIC_CONTINUATION)

NONZERO_DERIVATIVE
    -> supports(ANALYTIC_LOCAL_INVERSE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "analytic function" ~= "function locally represented by a convergent power series",
    "complex analytic function" ~= "holomorphic function",
    "Taylor expansion" ~= "local power-series representation using derivatives at the centre",
    "entire function" ~= "analytic function on the entire complex plane",
    "analytic continuation" ~= "extension of an analytic function while preserving agreement on overlaps",
    "order of zero" ~= "multiplicity of an isolated zero of a nonzero analytic function"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    analytic_function_requires_local_power_series_representation: true,
    complex_holomorphic_and_analytic_are_equivalent_on_open_domains: true,
    taylor_coefficients_are_uniquely_determined: true,
    zeros_of_nonzero_analytic_function_are_isolated_on_connected_domain: true,
    identity_theorem_requires_connected_domain_and_interior_accumulation_point: true,
    quotient_analyticity_requires_nonzero_denominator: true,
    analytic_continuation_may_encounter_singularities_or_monodromy: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    holomorphic_equivalence: 0.99,
    taylor_relation: 0.99,
    identity_zero_relation: 0.99,
    continuation_relation: 0.99,
    inverse_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_and_complex_domain)
    -> identify(point_or_region_of_interest)
    -> test(holomorphicity_or_power_series_representation)
    -> construct(taylor_coefficients_if_possible)
    -> determine(radius_of_convergence)
    -> analyse(zeros_and_multiplicities_if_requested)
    -> apply(identity_theorem_if_comparing_functions)
    -> trace(analytic_continuation_if_requested)
    -> classify(singularities_or_continuation_obstructions)
    -> apply(local_inverse_theorem_if_derivative_nonzero)
    -> validate(domain_connectedness_and_regularities)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed analytic-function semantic graph",
    primary_entity: ANALYTIC_FUNCTION,
    preserve: [
        function,
        complex_domain,
        expansion_centre,
        taylor_coefficients,
        radius_of_convergence,
        zeros,
        singularities,
        continuation_domain
    ],
    infer_through: [
        holomorphic_functions,
        power_series,
        cauchy_integral_formula,
        taylor_series,
        identity_theorem,
        zero_multiplicity,
        analytic_continuation,
        local_inverses
    ],
    avoid: [
        confusing_real_smoothness_with analyticity,
        assuming_power_series_converges_beyond its radius,
        ignoring_domain_connectedness_in identity theorem,
        treating_accumulation_point_outside_domain_as sufficient,
        ignoring_singularities_when estimating radius,
        assuming_global_inverse_from local nonzero derivative
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + ANALYTIC_FUNCTION
    + HOLOMORPHIC_FUNCTION
    + POWER_SERIES
    + TAYLOR_SERIES
    + RADIUS_OF_CONVERGENCE
    + IDENTITY_THEOREM
    + ANALYTIC_CONTINUATION
    + ENTIRE_FUNCTION
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> ANALYTIC_FUNCTION
    -> FUNCTION_DOMAIN_IDENTIFICATION
    -> ANALYTICITY_HOLOMORPHY_ANALYSIS
    -> TAYLOR_SERIES_CONSTRUCTION
    -> RADIUS_ANALYSIS
    -> ZERO_MULTIPLICITY_ANALYSIS
    -> IDENTITY_THEOREM_ANALYSIS
    -> CONTINUATION_ANALYSIS
    -> SINGULARITY_OBSTRUCTION_ANALYSIS
    -> LOCAL_INVERSE_ANALYSIS
    -> DOMAIN_HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW