Cauchy Integral Ontology

flow "CauchyIntegralOntology" v1.0

# ============================================================
# CAUCHY INTEGRAL ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/complex-analysis/cauchy-integral-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Complex Analysis"
ENTITY := "Cauchy Integral"

# ---------- CORE ENTITY ----------
CAUCHY_INTEGRAL := {
    type: ComplexIntegralTheory,
    domain: ComplexAnalysis,
    purpose: "Represent the Cauchy integral theorem and Cauchy integral formula, linking holomorphic functions, closed contour integrals, interior values and complex derivatives."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    CAUCHY_INTEGRAL,
    CAUCHY_INTEGRAL_THEOREM,
    CAUCHY_INTEGRAL_FORMULA,
    HOLOMORPHIC_FUNCTION,
    CONTOUR,
    CLOSED_CONTOUR,
    SIMPLE_CONNECTEDNESS,
    PRIMITIVE,
    PATH_INDEPENDENCE,
    WINDING_NUMBER,
    COMPLEX_DERIVATIVE,
    CAUCHY_DERIVATIVE_FORMULA,
    CAUCHY_ESTIMATE,
    ANALYTIC_FUNCTION
}

# ---------- CAUCHY INTEGRAL THEOREM ----------
let
    D
        := suitable_domain_in_C

and
    f
        -> HOLOMORPHIC_ON(D)

if
    gamma
        -> closed_piecewise_smooth_contour
       contractible_in(D)

then

closed_integral_gamma(
    f(z) dz
)
    = 0

CAUCHY_INTEGRAL_THEOREM
    -> VANISHING_CLOSED_CONTOUR_INTEGRAL

# ---------- SIMPLY CONNECTED FORM ----------
if
    D simply_connected
    AND
    f holomorphic_on(D)

then

for every closed contour gamma in D:

closed_integral_gamma(
    f(z) dz
)
    = 0

SIMPLY_CONNECTEDNESS
    -> removes(HOLES_AS_GLOBAL_OBSTRUCTIONS)

# ---------- PRIMITIVE RELATION ----------
if
    f holomorphic_on(simply_connected_domain_D)

then

exists
    F: D -> C

such_that

F_prime
    = f

PRIMITIVE
    -> implies(PATH_INDEPENDENCE)

# ---------- PATH INDEPENDENCE ----------
if
    gamma_1
    and
    gamma_2

have same endpoints
and lie in D

then

integral_gamma1(
    f(z) dz
)
=
integral_gamma2(
    f(z) dz
)

provided
    f has_primitive_on(D)

PATH_INDEPENDENCE
    -> equivalent_to(ZERO_CLOSED_CONTOUR_INTEGRALS)
       under(APPROPRIATE_DOMAIN_CONDITIONS)

# ---------- CAUCHY INTEGRAL FORMULA ----------
let
    C
        -> positively_oriented_simple_closed_contour

and
    f
        -> holomorphic_on_neighbourhood_of(C_AND_ITS_INTERIOR)

for
    z0 inside C

f(z0)
    =
1/(2*pi*i)
*
closed_integral_C(
    f(z)/(z-z0)
    dz
)

CAUCHY_INTEGRAL_FORMULA
    -> reconstructs(INTERIOR_VALUE_FROM_BOUNDARY_DATA)

# ---------- GENERAL WINDING-NUMBER FORM ----------
if
    gamma closed
    and
    z0 not_on(gamma)

and
    f holomorphic_on_relevant_region

then

closed_integral_gamma(
    f(z)/(z-z0)
    dz
)
=
2*pi*i
*
Ind(gamma,z0)
*
f(z0)

WINDING_NUMBER
    -> counts(ORIENTED_ENCIRCLEMENT)

# ---------- CAUCHY KERNEL ----------
CAUCHY_KERNEL
    :=
1/(z-z0)

CAUCHY_KERNEL
    -> has(SIMPLE_POLE_AT_z0)

RESIDUE_OF_KERNEL
    = 1

CAUCHY_INTEGRAL_FORMULA
    -> integral_reproduction_property_of(CAUCHY_KERNEL)

# ---------- HIGHER DERIVATIVES ----------
for
    n >= 0

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

CAUCHY_DERIVATIVE_FORMULA
    -> determines(ALL_COMPLEX_DERIVATIVES)

# ---------- HOLOMORPHIC IMPLIES SMOOTH ----------
CAUCHY_DERIVATIVE_FORMULA
    -> implies(
        derivatives_of_all_orders_exist
    )

HOLOMORPHIC_FUNCTION
    -> INFINITELY_COMPLEX_DIFFERENTIABLE

# ---------- HOLOMORPHIC IMPLIES ANALYTIC ----------
expand

1/(z-w)

as
    geometric_series
       for(|z-z0| < suitable_radius)

insert_into
    CAUCHY_INTEGRAL_FORMULA

-> derive(TAYLOR_SERIES)

therefore

HOLOMORPHIC
    -> ANALYTIC

CAUCHY_FORMULA
    -> foundational_to(POWER_SERIES_STRUCTURE)

# ---------- TAYLOR COEFFICIENTS ----------
a_n
    :=
f^(n)(z0)/n!

also

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

TAYLOR_COEFFICIENT
    -> boundary_integral_quantity

# ---------- CAUCHY ESTIMATES ----------
if
    f holomorphic
    on neighbourhood_of(
        closed_disk(z0,R)
    )

and

M
    :=
max_on_|z-z0|=R(
    |f(z)|
)

then

|f^(n)(z0)|
    <=
n!*M/R^n

CAUCHY_ESTIMATE
    -> controls(DERIVATIVE_MAGNITUDE)

# ---------- LIOUVILLE RELATION ----------
if
    f entire
    AND bounded

then

CAUCHY_ESTIMATES
    -> higher_derivatives_vanish_as(R -> infinity)

in particular

f_prime
    = 0

therefore

f
    -> CONSTANT

LIOUVILLE_THEOREM
    -> consequence_of(CAUCHY_ESTIMATES)

# ---------- FUNDAMENTAL THEOREM OF ALGEBRA ----------
LIOUVILLE_THEOREM
    -> supports(STANDARD_PROOF_OF_FUNDAMENTAL_THEOREM_OF_ALGEBRA)

CAUCHY_THEORY
    -> connects_to(POLYNOMIAL_ROOT_EXISTENCE)

# ---------- MEAN VALUE PROPERTY ----------
for holomorphic f:

f(z0)
    =
1/(2*pi)
*
integral_from_0_to_2pi(
    f(z0+R*e^(i*theta))
    dtheta
)

when
    closed_disk_inside_domain

MEAN_VALUE_PROPERTY
    -> consequence_of(CAUCHY_INTEGRAL_FORMULA)

# ---------- HARMONIC RELATION ----------
REAL_AND_IMAGINARY_PARTS_OF_HOLOMORPHIC_FUNCTION
    -> HARMONIC

CAUCHY_INTEGRAL_THEORY
    -> related_to(HARMONIC_MEAN_VALUE_PROPERTIES)

# ---------- MORERA THEOREM ----------
if
    f continuous_on(D)

and

closed_integral_triangle(
    f(z) dz
)
    = 0

for every triangle contained in D

then

f
    -> HOLOMORPHIC_ON(D)

MORERA_THEOREM
    -> partial_converse_to(CAUCHY_INTEGRAL_THEOREM)

# ---------- DOMAIN WITH HOLES ----------
if
    D not simply_connected

then

holomorphic f
    -/> necessarily_has_zero_integral_around_every_closed_contour

example:
    f(z)=1/z
    on C minus {0}

closed_integral_unit_circle(
    1/z dz
)
    =
2*pi*i

TOPOLOGICAL_HOLE
    -> prevents(GLOBAL_PRIMITIVE)

# ---------- HOMOTOPY FORM ----------
if
    gamma_0
    homotopic_to
    gamma_1
    within D

and
    f holomorphic_on(D)

then

integral_gamma0(f dz)
    =
integral_gamma1(f dz)

CAUCHY_THEORY
    -> contour_integrals_depend_on(HOMOTOPY_CLASS)
       when(SINGULARITIES_OR_HOLES_PRESENT)

# ---------- CONTOUR DEFORMATION ----------
CONTOUR
    -> may_be_deformed

without changing integral

if
    deformation_region_contains_no_singularity
    and integrand_holomorphic_throughout

CONTOUR_DEFORMATION
    -> practical_consequence_of(CAUCHY_THEORY)

# ---------- RESIDUE THEORY RELATION ----------
CAUCHY_INTEGRAL_FORMULA
    -> special_case_of(RESIDUE_THEOREM)

RESIDUE_THEOREM
    -> generalises(CAUCHY_KERNEL_CONTRIBUTIONS_TO_MULTIPLE_SINGULARITIES)

CAUCHY_THEORY
    -> foundation_of(RESIDUE_CALCULUS)

# ---------- APPLICATIONS ----------
CAUCHY_INTEGRAL
    -> applied_to(HOLOMORPHIC_FUNCTION_THEORY)
    -> applied_to(TAYLOR_SERIES)
    -> applied_to(RESIDUE_THEORY)
    -> applied_to(CONTOUR_INTEGRATION)
    -> applied_to(CONFORMAL_MAPPING)
    -> applied_to(HARMONIC_FUNCTIONS)
    -> applied_to(COMPLEX_DIFFERENTIAL_EQUATIONS)

# ---------- RELATION GRAPH ----------
HOLOMORPHIC_FUNCTION
    + CONTRACTIBLE_CLOSED_CONTOUR
    -> ZERO_CONTOUR_INTEGRAL

SIMPLY_CONNECTED_DOMAIN
    + HOLOMORPHIC_FUNCTION
    -> PRIMITIVE

CAUCHY_INTEGRAL_FORMULA
    -> determines(FUNCTION_VALUE)

CAUCHY_DERIVATIVE_FORMULA
    -> determines(HIGHER_DERIVATIVES)

CAUCHY_ESTIMATE
    -> bounds(DERIVATIVES)

CAUCHY_FORMULA
    -> implies(ANALYTICITY)

DOMAIN_TOPOLOGY
    -> controls(GLOBAL_INTEGRAL_BEHAVIOUR)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Cauchy integral theorem" ~= "vanishing of closed contour integrals of holomorphic functions under appropriate domain conditions",
    "Cauchy integral formula" ~= "boundary contour representation of an interior holomorphic function value",
    "Cauchy derivative formula" ~= "contour representation of higher derivatives of a holomorphic function",
    "Cauchy kernel" ~= "kernel 1/(z-z0) used to reproduce holomorphic values",
    "Cauchy estimate" ~= "derivative bound obtained from the Cauchy integral formula",
    "path independence" ~= "integral depends only on endpoints when a primitive exists"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    cauchy_integral_theorem_requires_holomorphy_and_valid_domain_conditions: true,
    simply_connected_form_requires_contour_inside_domain: true,
    cauchy_integral_formula_requires_z0_inside_contour_and_no_singularity_on_contour: true,
    contour_orientation_affects_sign: true,
    winding_number_required_for_general_closed_contours: true,
    domain_holes_can_prevent_global_primitive: true,
    cauchy_estimates_require_holomorphy_on_neighbourhood_of_closed_disk: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_theorem_structure: 0.99,
    primitive_path_relation: 0.99,
    integral_formula_relation: 0.99,
    derivative_formula_relation: 0.99,
    estimate_analyticity_relation: 0.99,
    topology_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_domain_and_contour)
    -> verify(holomorphy)
    -> inspect(domain_topology_and_singularities)
    -> determine(contractibility_or_winding_number)
    -> select(cauchy_theorem_or_integral_formula)
    -> verify(point_location_relative_to_contour)
    -> compute(function_value_or_derivative_if_requested)
    -> apply(cauchy_estimates_if_bounds_requested)
    -> infer(analyticity_or_primitive_properties_if_relevant)
    -> validate(contour_orientation_and_domain_conditions)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Cauchy-integral semantic graph",
    primary_entity: CAUCHY_INTEGRAL,
    preserve: [
        function,
        domain,
        contour,
        orientation,
        interior_point,
        holomorphy_status,
        winding_number,
        singularities
    ],
    infer_through: [
        primitives,
        path_independence,
        cauchy_integral_theorem,
        cauchy_integral_formula,
        derivative_formulas,
        cauchy_estimates,
        analyticity,
        contour_deformation
    ],
    avoid: [
        applying_cauchy_theorem_without domain conditions,
        ignoring_holes_or_singularities,
        using_integral_formula_for point_outside contour without winding adjustment,
        ignoring_orientation,
        assuming_holomorphic_function_has global primitive on arbitrary domain,
        applying_cauchy_estimate_without valid enclosing disk
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + CAUCHY_INTEGRAL
    + CAUCHY_INTEGRAL_THEOREM
    + CAUCHY_INTEGRAL_FORMULA
    + PRIMITIVE
    + PATH_INDEPENDENCE
    + WINDING_NUMBER
    + CAUCHY_DERIVATIVE_FORMULA
    + CAUCHY_ESTIMATE
    + MORERA_THEOREM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> CAUCHY_INTEGRAL
    -> FUNCTION_DOMAIN_CONTOUR_IDENTIFICATION
    -> HOLOMORPHY_VALIDATION
    -> TOPOLOGY_SINGULARITY_ANALYSIS
    -> CONTRACTIBILITY_WINDING_ANALYSIS
    -> THEOREM_SELECTION
    -> POINT_CONTOUR_RELATION_ANALYSIS
    -> VALUE_DERIVATIVE_COMPUTATION
    -> CAUCHY_ESTIMATE_ANALYSIS
    -> ANALYTICITY_PRIMITIVE_ANALYSIS
    -> ORIENTATION_DOMAIN_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW