Complex Derivative Ontology

flow "ComplexDerivativeOntology" v1.0

# ============================================================
# COMPLEX DERIVATIVE ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
COMPLEX_DERIVATIVE := {
    type: DerivativeConcept,
    domain: ComplexAnalysis,
    purpose: "Represent the derivative of a complex-valued function with respect to a complex variable, requiring the difference quotient to approach the same limit from every complex direction."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    COMPLEX_DERIVATIVE,
    COMPLEX_FUNCTION,
    COMPLEX_LIMIT,
    DIFFERENCE_QUOTIENT,
    COMPLEX_DIFFERENTIABILITY,
    HOLOMORPHIC_FUNCTION,
    CAUCHY_RIEMANN_EQUATIONS,
    REAL_PART,
    IMAGINARY_PART,
    DIRECTION_INDEPENDENCE,
    CHAIN_RULE,
    POWER_SERIES,
    ANALYTIC_FUNCTION,
    CONFORMAL_MAPPING
}

# ---------- BASIC DEFINITION ----------
let
    f: D subset_of C -> C

and
    z0 in D

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

if
    limit_exists_in(C)

COMPLEX_DERIVATIVE
    -> LIMIT_OF(COMPLEX_DIFFERENCE_QUOTIENT)

# ---------- DOMAIN REQUIREMENT ----------
for derivative_at(z0):

z0
    -> accumulation_point_of(DOMAIN)

usually
    z0 -> interior_point_of(D)
       for(HOLOMORPHIC_CONTEXT)

DIFFERENCE_QUOTIENT
    -> requires(z0+h in DOMAIN)

# ---------- DIRECTION INDEPENDENCE ----------
h
    -> complex_increment

h
    -> may_approach(0_FROM_ANY_DIRECTION)

COMPLEX_DERIVATIVE_EXISTS
    iff
    SAME_LIMIT_FOR_ALL_COMPLEX_APPROACHES

DIRECTION_INDEPENDENCE
    -> central_to(COMPLEX_DIFFERENTIABILITY)

# ---------- REAL / IMAGINARY DECOMPOSITION ----------
write

z
    = x+i*y

and

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

u
    -> REAL_PART_FUNCTION

v
    -> IMAGINARY_PART_FUNCTION

COMPLEX_DERIVATIVE
    -> constrains(PARTIAL_DERIVATIVES_OF_u_v)

# ---------- REAL DIRECTION ----------
take
    h = delta_x
    real

then

f_prime(z0)
    =
u_x
+
i*v_x

provided
    relevant_partials_exist

# ---------- IMAGINARY DIRECTION ----------
take
    h = i*delta_y

then

f_prime(z0)
    =
v_y
-
i*u_y

provided
    relevant_partials_exist

# ---------- CAUCHY-RIEMANN EQUATIONS ----------
equating
    REAL_DIRECTION_DERIVATIVE
    and
    IMAGINARY_DIRECTION_DERIVATIVE

gives

u_x
    = v_y

u_y
    = -v_x

CAUCHY_RIEMANN_EQUATIONS
    -> necessary_for(COMPLEX_DIFFERENTIABILITY)
       under(PARTIAL_DERIVATIVE_EXISTENCE)

# ---------- SUFFICIENCY ----------
if
    u,v
        -> have_continuous_first_partial_derivatives
       near(z0)

and

CAUCHY_RIEMANN_EQUATIONS
    -> hold_near(z0)

then

f
    -> COMPLEX_DIFFERENTIABLE_NEAR(z0)

CR_EQUATIONS_ALONE_AT_SINGLE_POINT
    -/> generally_sufficient_without(REGULARITY)

# ---------- COMPLEX VS REAL DIFFERENTIABILITY ----------
f: C -> C
    -> may_be_viewed_as(F: R^2 -> R^2)

REAL_DIFFERENTIABILITY
    -/> implies(COMPLEX_DIFFERENTIABILITY)

COMPLEX_DIFFERENTIABILITY
    -> real_differentiability
       under(STANDARD_LOCAL_REGULARITY_IMPLIED_BY_EXISTENCE_OF_COMPLEX_DERIVATIVE_AT_POINT)

COMPLEX_LINEARITY_OF_REAL_DERIVATIVE
    -> equivalent_to(CAUCHY_RIEMANN_STRUCTURE)

# ---------- DERIVATIVE AS LOCAL LINEAR MAP ----------
if
    f complex_differentiable_at(z0)

then

f(z0+h)
    =
f(z0)
+
f_prime(z0)*h
+
o(|h|)

MULTIPLICATION_BY(f_prime(z0))
    -> LOCAL_COMPLEX_LINEAR_APPROXIMATION

# ---------- GEOMETRIC INTERPRETATION ----------
write

f_prime(z0)
    =
rho*e^(i*theta)

if
    f_prime(z0) != 0

then locally:

rho
    -> SCALE_FACTOR

theta
    -> ROTATION_ANGLE

COMPLEX_DERIVATIVE
    -> local_rotation_and_scaling

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

f
    -> CONFORMAL_AT(z0)

NONZERO_COMPLEX_DERIVATIVE
    -> preserves(ANGLES_LOCALLY)

# ---------- ZERO DERIVATIVE ----------
if
    f_prime(z0)=0

then

FIRST_ORDER_LOCAL_LINEAR_TERM
    -> vanishes

CONFORMALITY
    -> not_guaranteed_at(z0)

higher_order_terms
    -> determine(LOCAL_BEHAVIOUR)

# ---------- DERIVATIVE RULES ----------
if
    f,g complex_differentiable_at(z0)

then

(f+g)'(z0)
    =
f'(z0)+g'(z0)

(alpha*f)'(z0)
    =
alpha*f'(z0)

(f*g)'(z0)
    =
f'(z0)g(z0)
+
f(z0)g'(z0)

# ---------- QUOTIENT RULE ----------
if
    g(z0) != 0

then

(f/g)'(z0)
    =
[
    f'(z0)g(z0)
    -
    f(z0)g'(z0)
]
/
g(z0)^2

QUOTIENT_RULE
    -> requires(NONZERO_DENOMINATOR)

# ---------- CHAIN RULE ----------
if
    f complex_differentiable_at(z0)

and
    g complex_differentiable_at(f(z0))

then

(g composed_with f)'(z0)
    =
g'(f(z0))
*
f'(z0)

CHAIN_RULE
    -> preserves(COMPLEX_DIFFERENTIABILITY_UNDER_COMPOSITION)

# ---------- INVERSE DERIVATIVE ----------
if
    f holomorphic_near(z0)
    AND
    f_prime(z0) != 0
    AND local_inverse_exists

then

(f_inverse)'(f(z0))
    =
1/f_prime(z0)

COMPLEX_INVERSE_FUNCTION_THEOREM
    -> supplies(LOCAL_HOLOMORPHIC_INVERSE)

# ---------- ELEMENTARY DERIVATIVES ----------
d/dz(z^n)
    =
n*z^(n-1)

d/dz(exp(z))
    =
exp(z)

d/dz(sin(z))
    =
cos(z)

d/dz(cos(z))
    =
-sin(z)

ELEMENTARY_COMPLEX_FUNCTIONS
    -> obey(FAMILIAR_DERIVATIVE_RULES)

# ---------- HOLOMORPHICITY ----------
if
    f complex_differentiable
    at every point of open set D

then

f
    -> HOLOMORPHIC_ON(D)

COMPLEX_DERIVATIVE
    -> local_building_block_of(HOLOMORPHICITY)

# ---------- ANALYTICITY ----------
if
    f holomorphic_on(D)

then

f
    -> ANALYTIC_ON(D)

f
    -> has(DERIVATIVES_OF_ALL_ORDERS)

COMPLEX_DIFFERENTIABILITY_ON_OPEN_SET
    -> unexpectedly_strong_regular_structure

# ---------- CAUCHY DERIVATIVE FORMULA ----------
if
    f holomorphic
    on neighbourhood_of(region_inside_C)

then

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

COMPLEX_DERIVATIVES
    -> determined_by(CONTOUR_VALUES)

# ---------- POWER SERIES DERIVATIVE ----------
if

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

inside radius R

then

f_prime(z)
    =
sum_from_n_1_to_infinity(
    n*a_n*(z-z0)^(n-1)
)

TERM_BY_TERM_DIFFERENTIATION
    -> valid_inside(RADIUS_OF_CONVERGENCE)

# ---------- NONEXAMPLE: CONJUGATION ----------
f(z)
    := conjugate(z)

write

u=x
v=-y

u_x=1
v_y=-1

CAUCHY_RIEMANN
    -> FAIL

therefore

conjugation_function
    -> nowhere_complex_differentiable

# ---------- NONEXAMPLE: MODULUS SQUARED ----------
f(z)
    := |z|^2
       = x^2+y^2

as complex-valued real output:

u=x^2+y^2
v=0

CR_EQUATIONS
    -> hold_only_at(0)

f
    -> complex_differentiable_at(0)
    with
    f_prime(0)=0

but

f
    -/> holomorphic_on_any_neighbourhood_of(0)

POINTWISE_COMPLEX_DIFFERENTIABILITY
    -/> HOLOMORPHY

# ---------- APPLICATIONS ----------
COMPLEX_DERIVATIVE
    -> applied_to(HOLOMORPHIC_FUNCTIONS)
    -> applied_to(CONFORMAL_MAPPING)
    -> applied_to(POWER_SERIES)
    -> applied_to(COMPLEX_DYNAMICS)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
COMPLEX_LIMIT
    -> defines(COMPLEX_DERIVATIVE)

COMPLEX_DERIVATIVE
    -> requires(DIRECTION_INDEPENDENCE)

COMPLEX_DERIVATIVE
    -> implies(CAUCHY_RIEMANN_AT_REGULAR_POINTS)

COMPLEX_DIFFERENTIABILITY_ON_OPEN_SET
    -> HOLOMORPHIC_FUNCTION

HOLOMORPHIC_FUNCTION
    -> ANALYTIC_FUNCTION

NONZERO_COMPLEX_DERIVATIVE
    -> supports(CONFORMALITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "complex derivative" ~= "direction-independent limit of the complex difference quotient",
    "complex differentiability" ~= "existence of a complex derivative at a point",
    "holomorphic derivative" ~= "complex derivative of a holomorphic function",
    "Cauchy-Riemann conditions" ~= "component derivative relations required by complex differentiability",
    "local complex-linear approximation" ~= "first-order approximation given by multiplication by the complex derivative",
    "nonzero complex derivative" ~= "local rotation-scaling factor for a holomorphic map"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    complex_difference_quotient_limit_must_be_direction_independent: true,
    real_differentiability_does_not_imply_complex_differentiability: true,
    cauchy_riemann_equations_at_single_point_are_not_generally_sufficient: true,
    holomorphicity_requires_complex_differentiability_on_open_set: true,
    conformality_requires_nonzero_derivative: true,
    quotient_rule_requires_nonzero_denominator: true,
    branch_dependent_functions_require_domain_and_branch_handling: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    direction_independence: 0.99,
    cauchy_riemann_relation: 0.99,
    local_linear_relation: 0.99,
    holomorphic_analytic_relation: 0.99,
    conformal_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(complex_function_and_point)
    -> verify(domain_accumulation_structure)
    -> construct(complex_difference_quotient)
    -> test(limit_along_multiple_complex_directions)
    -> decompose_into(real_and_imaginary_components_if_useful)
    -> test(cauchy_riemann_conditions)
    -> verify(component_regularities_for_sufficiency)
    -> determine(complex_differentiability)
    -> classify(holomorphicity_if_open_neighbourhood)
    -> apply(derivative_rules_or_power_series_methods_if_requested)
    -> analyse(local_conformal_structure_if_derivative_nonzero)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed complex-derivative semantic graph",
    primary_entity: COMPLEX_DERIVATIVE,
    preserve: [
        function,
        domain,
        evaluation_point,
        complex_increment,
        difference_quotient,
        real_component,
        imaginary_component,
        derivative_value
    ],
    infer_through: [
        complex_limits,
        cauchy_riemann_equations,
        local_linear_approximations,
        holomorphic_functions,
        analytic_functions,
        derivative_rules,
        conformal_maps
    ],
    avoid: [
        checking_derivative_along_only_one path,
        treating_real_partial_derivatives_as sufficient automatically,
        assuming_cauchy_riemann_at_one_point proves holomorphy,
        confusing_pointwise_complex_differentiability_with neighbourhood holomorphy,
        claiming_conformality_when derivative_is_zero,
        ignoring_function_domain
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + COMPLEX_DERIVATIVE
    + COMPLEX_LIMIT
    + DIFFERENCE_QUOTIENT
    + DIRECTION_INDEPENDENCE
    + CAUCHY_RIEMANN_EQUATIONS
    + HOLOMORPHIC_FUNCTION
    + ANALYTIC_FUNCTION
    + CONFORMAL_MAPPING
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> COMPLEX_DERIVATIVE
    -> FUNCTION_POINT_IDENTIFICATION
    -> DOMAIN_VALIDATION
    -> DIFFERENCE_QUOTIENT_CONSTRUCTION
    -> DIRECTIONAL_LIMIT_ANALYSIS
    -> COMPONENT_DECOMPOSITION
    -> CAUCHY_RIEMANN_ANALYSIS
    -> REGULARITY_VALIDATION
    -> DIFFERENTIABILITY_CLASSIFICATION
    -> HOLOMORPHY_ANALYSIS
    -> DERIVATIVE_RULE_ANALYSIS
    -> CONFORMALITY_ANALYSIS
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW