Holomorphic Function Ontology

flow "HolomorphicFunctionOntology" v1.0

# ============================================================
# HOLOMORPHIC FUNCTION ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
HOLOMORPHIC_FUNCTION := {
    type: ComplexFunctionProperty,
    domain: ComplexAnalysis,
    purpose: "Represent a complex-valued function that is complex differentiable at every point of an open domain, implying analytic power-series structure and strong integral and geometric properties."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    HOLOMORPHIC_FUNCTION,
    COMPLEX_FUNCTION,
    OPEN_DOMAIN,
    COMPLEX_DERIVATIVE,
    ANALYTIC_FUNCTION,
    CAUCHY_RIEMANN_EQUATIONS,
    CAUCHY_INTEGRAL_THEOREM,
    CAUCHY_INTEGRAL_FORMULA,
    POWER_SERIES,
    HARMONIC_FUNCTION,
    ENTIRE_FUNCTION,
    ZERO,
    IDENTITY_THEOREM,
    CONFORMAL_MAPPING,
    MAXIMUM_MODULUS_PRINCIPLE
}

# ---------- BASIC DEFINITION ----------
let
    D
        := OPEN_SUBSET_OF(C)

and
    f: D -> C

f
    -> HOLOMORPHIC_ON(D)

iff

for_every
    z0 in D

f_prime(z0)
    :=
lim_(h->0)
(
    f(z0+h)-f(z0)
)
/h

exists

HOLOMORPHY
    -> COMPLEX_DIFFERENTIABILITY_AT_EVERY_DOMAIN_POINT

# ---------- LOCAL PROPERTY ----------
f
    -> holomorphic_at(z0)

if

exists neighbourhood U of z0
such_that

f
    -> complex_differentiable_on(U)

HOLOMORPHY
    -> LOCAL_PROPERTY

# ---------- COMPLEX DIFFERENTIABILITY ----------
COMPLEX_DERIVATIVE
    -> requires(LIMIT_INDEPENDENT_OF_DIRECTION)

h
    -> approaches_zero_from(ANY_COMPLEX_DIRECTION)

COMPLEX_DIFFERENTIABILITY
    -> stronger_than(REAL_DIFFERENTIABILITY)

# ---------- COMPONENT FORM ----------
write

f(z)
    =
u(x,y)
+
i*v(x,y)

where

z=x+i*y

u
    -> REAL_PART_FUNCTION

v
    -> IMAGINARY_PART_FUNCTION

# ---------- CAUCHY-RIEMANN ----------
if
    f complex_differentiable_at(z0)
    AND component_partials_exist
then

u_x
    = v_y

u_y
    = -v_x

CAUCHY_RIEMANN_EQUATIONS
    -> necessary_conditions

if
    u,v have_continuous_first_partials_near(z0)
    AND CR_EQUATIONS_hold
then

f
    -> holomorphic_near(z0)

# ---------- DERIVATIVE FORMULA FROM COMPONENTS ----------
under holomorphy:

f_prime(z)
    =
u_x + i*v_x

also

f_prime(z)
    =
v_y - i*u_y

CAUCHY_RIEMANN
    -> ensures(CONSISTENCY_OF_DIRECTIONAL_DIFFERENCE_QUOTIENTS)

# ---------- ANALYTICITY ----------
HOLOMORPHIC_FUNCTION
    -> ANALYTIC_FUNCTION

for each z0 in D:

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

in some neighbourhood

HOLOMORPHIC
    <-> ANALYTIC
       in(COMPLEX_ANALYSIS)

# ---------- INFINITE DIFFERENTIABILITY ----------
f holomorphic
    -> f_prime holomorphic
    -> f_double_prime holomorphic
    -> derivatives_of_all_orders_exist

HOLOMORPHIC_FUNCTION
    -> C_INFINITY_IN_COMPLEX_SENSE

# ---------- CAUCHY INTEGRAL THEOREM ----------
if
    f holomorphic
    on simply_connected_domain_D

and
    gamma closed_piecewise_smooth_contour_in(D)

then

closed_integral_gamma(
    f(z) dz
)
    = 0

HOLOMORPHY
    + DOMAIN_TOPOLOGY
    -> VANISHING_CLOSED_CONTOUR_INTEGRALS

# ---------- PRIMITIVE ----------
if
    D simply_connected
    AND f holomorphic_on(D)
then

exists
    F holomorphic_on(D)

such_that

F_prime
    = f

PRIMITIVE
    -> path_independence_of(CONTOUR_INTEGRAL)

# ---------- CAUCHY INTEGRAL FORMULA ----------
if
    f holomorphic
    on neighbourhood_of(region_inside_and_on_C)

then

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

for
    z0 inside C

CAUCHY_FORMULA
    -> determines(INTERIOR_VALUES_FROM_BOUNDARY_DATA)

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

CAUCHY_DERIVATIVE_FORMULA
    -> implies(INFINITE_DIFFERENTIABILITY)

# ---------- CAUCHY ESTIMATES ----------
if
    |f(z)| <= M
    on circle
    |z-z0|=R

then

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

CAUCHY_ESTIMATES
    -> bound(DERIVATIVES_BY_FUNCTION_SIZE)

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

iff

f holomorphic_on(C)

ENTIRE
    -> global_holomorphy

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

# ---------- HARMONIC COMPONENTS ----------
if
    f=u+iv holomorphic
then

u_xx + u_yy
    = 0

v_xx + v_yy
    = 0

u,v
    -> HARMONIC

u
    and v
    -> HARMONIC_CONJUGATES_LOCALLY

# ---------- ZERO STRUCTURE ----------
if
    f holomorphic_on(connected_domain)
    AND
    f not_identically_zero
then

zeros_of_f
    -> ISOLATED

for zero z0:

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

where

g(z0) != 0

m
    -> ZERO_MULTIPLICITY

# ---------- IDENTITY THEOREM ----------
if
    f,g holomorphic_on(connected_domain_D)

and

f(z)=g(z)
    on set_with_accumulation_point_in(D)

then

f
    = g
    on D

HOLOMORPHIC_FUNCTIONS
    -> uniquely_determined_by(LOCAL_ACCUMULATING_DATA)

# ---------- MAXIMUM MODULUS ----------
if
    f nonconstant_holomorphic
    on connected_open_domain
then

|f|
    -> cannot_attain_local_maximum_in_interior

MAXIMUM_MODULUS_PRINCIPLE
    -> rigidity_of(HOLOMORPHIC_FUNCTION)

# ---------- MINIMUM MODULUS ----------
if
    f holomorphic
    AND nonvanishing
then

1/f
    -> holomorphic

MAXIMUM_MODULUS_FOR(1/f)
    -> yields(MINIMUM_MODULUS_RESULTS)

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

f
    -> CONSTANT

LIOUVILLE_THEOREM
    -> consequence_of(CAUCHY_ESTIMATES)

# ---------- OPEN MAPPING ----------
if
    f nonconstant_holomorphic
    on domain
then

f
    -> OPEN_MAP

OPEN_MAPPING_THEOREM
    -> maps(OPEN_SETS_TO_OPEN_SETS)

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

f
    -> locally_bijective_near(z0)

LOCAL_INVERSE
    -> HOLOMORPHIC

COMPLEX_INVERSE_FUNCTION_THEOREM
    -> local_biholomorphic_structure

# ---------- CONFORMALITY ----------
if
    f holomorphic_at(z0)
    AND
    f_prime(z0) != 0
then

f
    -> CONFORMAL_AT(z0)

CONFORMAL
    -> preserves(ANGLES_AND_ORIENTATION_LOCALLY)

# ---------- ALGEBRAIC CLOSURE ----------
if
    f,g holomorphic_on(D)

then

f+g
    -> holomorphic

f-g
    -> holomorphic

f*g
    -> holomorphic

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

HOLOMORPHIC_FUNCTIONS
    -> form(ALGEBRA_UNDER_POINTWISE_OPERATIONS)

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

g composed_with f
    -> holomorphic_on(D)

CHAIN_RULE:
    (g composed_with f)'(z)
    =
g'(f(z))*f'(z)

# ---------- SINGULARITY RELATION ----------
if
    f holomorphic_on(
        punctured_neighbourhood_of(z0)
    )

but not holomorphic_at(z0)

then

z0
    -> ISOLATED_SINGULARITY

HOLOMORPHY
    -> baseline_against_which(SINGULARITIES_ARE_DEFINED)

# ---------- APPLICATIONS ----------
HOLOMORPHIC_FUNCTION
    -> applied_to(CONTOUR_INTEGRATION)
    -> applied_to(CONFORMAL_MAPPING)
    -> applied_to(HARMONIC_ANALYSIS)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(FLUID_DYNAMICS)
    -> applied_to(ELECTROMAGNETISM)
    -> applied_to(NUMBER_THEORY)

# ---------- RELATION GRAPH ----------
COMPLEX_DIFFERENTIABILITY
    -> on_open_domain_defines(HOLOMORPHY)

CAUCHY_RIEMANN_EQUATIONS
    -> support(HOLOMORPHY_TEST)

HOLOMORPHY
    -> implies(ANALYTICITY)

HOLOMORPHY
    -> enables(CAUCHY_INTEGRAL_RESULTS)

HOLOMORPHIC_COMPONENTS
    -> HARMONIC

HOLOMORPHY
    + NONZERO_DERIVATIVE
    -> CONFORMALITY

HOLOMORPHIC_ZERO_SET
    -> ISOLATED_UNLESS_IDENTICALLY_ZERO

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "holomorphic function" ~= "complex function differentiable at every point of an open domain",
    "complex differentiable function on an open set" ~= "holomorphic function",
    "analytic complex function" ~= "holomorphic function",
    "entire function" ~= "holomorphic function on the whole complex plane",
    "biholomorphic local map" ~= "locally invertible holomorphic map with holomorphic inverse",
    "harmonic conjugates" ~= "real and imaginary components linked by the Cauchy-Riemann equations"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    holomorphicity_requires_complex_differentiability_on_open_domain: true,
    complex_derivative_requires_direction_independent_limit: true,
    cauchy_riemann_sufficiency_requires_regularities: true,
    holomorphic_implies_analytic_in_complex_analysis: true,
    zeros_are_isolated_only_for_nonidentically_zero_function_on_connected_domain: true,
    conformality_requires_nonzero_derivative: true,
    global_contour_results_require_domain_topology_conditions: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    cauchy_riemann_relation: 0.99,
    analytic_equivalence: 0.99,
    cauchy_integral_relation: 0.99,
    zero_identity_relation: 0.99,
    conformal_inverse_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(complex_function_and_domain)
    -> verify(domain_is_open_for_holomorphy)
    -> test(complex_derivative)
    -> decompose_into(u_v_if_useful)
    -> apply(cauchy_riemann_with_regularities)
    -> classify(holomorphic_or_not)
    -> infer(analyticity_and_higher_derivatives_if_holomorphic)
    -> apply(cauchy_integral_theorems_if_domain_conditions_hold)
    -> analyse(zeros_identity_or_maximum_modulus_if_requested)
    -> test(nonzero_derivative_for_conformal_or_inverse_properties)
    -> validate(domain_connectedness_and_topology)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed holomorphic-function semantic graph",
    primary_entity: HOLOMORPHIC_FUNCTION,
    preserve: [
        function,
        domain,
        complex_derivative,
        real_component,
        imaginary_component,
        zero_set,
        derivative_nonzero_status,
        domain_topology
    ],
    infer_through: [
        cauchy_riemann_equations,
        analytic_functions,
        power_series,
        cauchy_integral_formula,
        harmonic_functions,
        identity_theorem,
        maximum_modulus,
        conformal_maps
    ],
    avoid: [
        treating_real_differentiability_as holomorphy,
        ignoring_open_domain requirement,
        using_cauchy_riemann_as sufficient without regularity,
        claiming_zero_isolation_for identically zero function,
        claiming_conformality_when derivative_zero,
        applying_global_cauchy_results_without topology conditions
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + HOLOMORPHIC_FUNCTION
    + COMPLEX_DERIVATIVE
    + CAUCHY_RIEMANN_EQUATIONS
    + ANALYTIC_FUNCTION
    + CAUCHY_INTEGRAL_FORMULA
    + HARMONIC_FUNCTION
    + IDENTITY_THEOREM
    + MAXIMUM_MODULUS_PRINCIPLE
    + CONFORMAL_MAPPING
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> HOLOMORPHIC_FUNCTION
    -> FUNCTION_DOMAIN_IDENTIFICATION
    -> OPEN_DOMAIN_VALIDATION
    -> COMPLEX_DERIVATIVE_ANALYSIS
    -> COMPONENT_CR_ANALYSIS
    -> HOLOMORPHY_CLASSIFICATION
    -> ANALYTICITY_DERIVATIVE_ANALYSIS
    -> CAUCHY_THEOREM_ANALYSIS
    -> ZERO_IDENTITY_MAXIMUM_ANALYSIS
    -> CONFORMAL_INVERSE_ANALYSIS
    -> DOMAIN_TOPOLOGY_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW