Complex Function Ontology

flow "ComplexFunctionOntology" v1.0

# ============================================================
# COMPLEX FUNCTION ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
COMPLEX_FUNCTION := {
    type: MathematicalFunction,
    domain: ComplexAnalysis,
    purpose: "Represent a mapping whose domain, codomain, or both involve complex numbers, with particular emphasis on complex-valued functions of a complex variable."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    COMPLEX_FUNCTION,
    COMPLEX_NUMBER,
    DOMAIN,
    CODOMAIN,
    REAL_PART,
    IMAGINARY_PART,
    MODULUS,
    ARGUMENT,
    COMPLEX_LIMIT,
    CONTINUITY,
    COMPLEX_DERIVATIVE,
    HOLOMORPHIC_FUNCTION,
    ANALYTIC_FUNCTION,
    CAUCHY_RIEMANN_EQUATIONS,
    COMPOSITION
}

# ---------- BASIC DEFINITION ----------
COMPLEX_FUNCTION
    := f: D -> C

where
    D subset_of C

z
    -> INPUT_COMPLEX_NUMBER

f(z)
    -> OUTPUT_COMPLEX_NUMBER

# ---------- INPUT REPRESENTATION ----------
z
    := x + i*y

x
    -> Re(z)

y
    -> Im(z)

COMPLEX_INPUT
    <-> point(x,y)_in_R2

# ---------- OUTPUT DECOMPOSITION ----------
f(z)
    =
u(x,y)
+
i*v(x,y)

u(x,y)
    := Re(f(z))

v(x,y)
    := Im(f(z))

COMPLEX_FUNCTION
    -> equivalent_to(PAIR_OF_REAL_VALUED_FUNCTIONS)
       on(SUBSET_OF_R2)

# ---------- DOMAIN ----------
DOMAIN(f)
    := D subset_of C

DOMAIN
    -> may_be(OPEN)
    -> may_be(CLOSED)
    -> may_be(CONNECTED)
    -> may_be(SIMPLY_CONNECTED)

DOMAIN_TOPOLOGY
    -> affects(COMPLEX_ANALYTIC_PROPERTIES)

# ---------- CODOMAIN / RANGE ----------
CODOMAIN(f)
    := C

RANGE(f)
    := {
        f(z) :
        z in DOMAIN(f)
    }

RANGE
    -> subset_of(CODOMAIN)

# ---------- CONSTANT FUNCTION ----------
f(z)
    := c

where
    c in C

CONSTANT_FUNCTION
    -> HOLOMORPHIC_EVERYWHERE

f_prime(z)
    = 0

# ---------- IDENTITY FUNCTION ----------
f(z)
    := z

IDENTITY_COMPLEX_FUNCTION
    -> ENTIRE

f_prime(z)
    = 1

# ---------- POLYNOMIAL FUNCTION ----------
P(z)
    :=
a_0
+
a_1*z
+
...
+
a_n*z^n

POLYNOMIAL
    -> ENTIRE

FUNDAMENTAL_THEOREM_OF_ALGEBRA
    -> nonconstant_complex_polynomial_has(ROOT_IN_C)

# ---------- RATIONAL FUNCTION ----------
R(z)
    :=
P(z)/Q(z)

DOMAIN
    := {
        z in C :
        Q(z) != 0
    }

RATIONAL_FUNCTION
    -> HOLOMORPHIC_ON(DOMAIN)
    -> MEROMORPHIC_ON(C)

zeros_of_Q
    -> possible_POLES_OR_REMOVABLE_POINTS
       depending_on(CANCELLATION)

# ---------- COMPLEX EXPONENTIAL ----------
exp(z)
    :=
e^x(
    cos(y)
    +
    i*sin(y)
)

EXPONENTIAL_FUNCTION
    -> ENTIRE

exp(z+w)
    =
exp(z)*exp(w)

exp(z)
    != 0

# ---------- COMPLEX TRIGONOMETRIC FUNCTIONS ----------
sin(z)
    :=
(
    e^(iz)-e^(-iz)
)/(2i)

cos(z)
    :=
(
    e^(iz)+e^(-iz)
)/2

sin(z)
    -> ENTIRE

cos(z)
    -> ENTIRE

# ---------- COMPLEX LOGARITHM ----------
log(z)
    :=
ln|z|
+
i*arg(z)

ARGUMENT
    -> multivalued_modulo(2*pi)

COMPLEX_LOGARITHM
    -> MULTIVALUED_WITHOUT(BRANCH)

BRANCH_OF_LOG
    -> single_valued_holomorphic_on(SUITABLE_CUT_DOMAIN)

# ---------- COMPLEX POWER ----------
z^alpha
    :=
exp(
    alpha*Log(z)
)

COMPLEX_POWER
    -> depends_on(LOGARITHM_BRANCH)
       unless(SPECIAL_INTEGER_EXPONENT_CONTEXT)

# ---------- MODULUS OF FUNCTION ----------
|f(z)|
    :=
sqrt(
    Re(f(z))^2
    +
    Im(f(z))^2
)

MODULUS
    -> REAL_NONNEGATIVE_FUNCTION

MAXIMUM_MODULUS_PRINCIPLE
    -> constrains(HOLOMORPHIC_FUNCTION_MODULUS)

# ---------- ARGUMENT OF FUNCTION ----------
arg(f(z))
    -> phase_of(COMPLEX_OUTPUT)

ARGUMENT
    -> undefined_when(f(z)=0)
    -> multivalued_modulo(2*pi)

PHASE
    -> relevant_to(CONFORMAL_AND_WAVE_INTERPRETATIONS)

# ---------- COMPLEX LIMIT ----------
lim_(z->z0) f(z)
    = L

iff

for_every epsilon > 0
exists delta > 0
such_that

0 < |z-z0| < delta
    -> |f(z)-L| < epsilon

LIMIT
    -> must_agree_along(ALL_PATHS_TO_z0)

# ---------- CONTINUITY ----------
f
    -> continuous_at(z0)

iff

lim_(z->z0) f(z)
    = f(z0)

equivalently

u,v
    -> continuous_at((x0,y0))
       under(STANDARD_COMPONENT_REPRESENTATION)

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

COMPLEX_DERIVATIVE
    -> requires(DIRECTION_INDEPENDENT_LIMIT)

COMPLEX_DIFFERENTIABILITY
    -> stronger_than(REAL_DIFFERENTIABILITY_OF_u_v)

# ---------- CAUCHY-RIEMANN ----------
if
    f=u+iv
    complex_differentiable
then

u_x
    = v_y

u_y
    = -v_x

CAUCHY_RIEMANN_EQUATIONS
    -> necessary_conditions

with suitable continuous first partials:

CR_EQUATIONS
    -> sufficient_for(LOCAL_HOLOMORPHY)

# ---------- HOLOMORPHIC FUNCTION ----------
f
    -> HOLOMORPHIC_ON(D)

if

D open
AND
f complex_differentiable_at_every_point_of(D)

HOLOMORPHIC
    -> implies(ANALYTIC)

# ---------- ANALYTIC FUNCTION ----------
f
    -> ANALYTIC_AT(z0)

if

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

in neighbourhood_of(z0)

ANALYTIC
    -> locally_power_series_representable

# ---------- COMPOSITION ----------
if
    f: D -> E
    AND
    g: E -> C

then

g composed_with f
    -> COMPLEX_FUNCTION

if
    f and g holomorphic
then

g composed_with f
    -> HOLOMORPHIC

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

# ---------- ALGEBRA ----------
if
    f,g holomorphic_on(D)

then

f+g
    -> holomorphic

f*g
    -> holomorphic

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

HOLOMORPHIC_FUNCTIONS
    -> closed_under(STANDARD_ALGEBRAIC_OPERATIONS_WHERE_DEFINED)

# ---------- ZEROS ----------
z0
    -> ZERO_OF(f)

if

f(z0)
    = 0

if
    f holomorphic
    AND not_identically_zero
then

zeros
    -> ISOLATED
       within(CONNECTED_DOMAIN)

ZERO_ORDER
    -> multiplicity_of_zero

# ---------- SINGULARITIES ----------
if
    f fails_to_be_holomorphic_at(z0)
    but holomorphic_in_punctured_neighbourhood
then

z0
    -> ISOLATED_SINGULARITY

ISOLATED_SINGULARITY
    -> REMOVABLE
    -> POLE
    -> ESSENTIAL

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

f
    -> LOCALLY_INVERTIBLE

LOCAL_INVERSE
    -> HOLOMORPHIC

COMPLEX_INVERSE_FUNCTION_THEOREM
    -> supports(LOCAL_CONFORMAL_STRUCTURE)

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

f
    -> CONFORMAL_AT(z0)

COMPLEX_FUNCTION
    -> may_preserve(ANGLES_LOCALLY)

# ---------- APPLICATIONS ----------
COMPLEX_FUNCTION
    -> applied_to(COMPLEX_ANALYSIS)
    -> applied_to(CONTOUR_INTEGRATION)
    -> applied_to(CONFORMAL_MAPPING)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(SIGNAL_PROCESSING)
    -> applied_to(FLUID_DYNAMICS)
    -> applied_to(ELECTROMAGNETISM)

# ---------- RELATION GRAPH ----------
COMPLEX_NUMBER
    -> input_of(COMPLEX_FUNCTION)

COMPLEX_FUNCTION
    -> decomposes_into(REAL_PART, IMAGINARY_PART)

COMPLEX_LIMIT
    -> defines(CONTINUITY)

COMPLEX_LIMIT
    -> defines(COMPLEX_DERIVATIVE)

COMPLEX_DIFFERENTIABILITY_ON_OPEN_DOMAIN
    -> HOLOMORPHIC_FUNCTION

HOLOMORPHIC_FUNCTION
    -> ANALYTIC_FUNCTION

NONZERO_DERIVATIVE
    -> supports(CONFORMALITY_AND_LOCAL_INVERSE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "complex function" ~= "function whose inputs or outputs involve complex numbers",
    "complex-valued function of a complex variable" ~= "mapping from a subset of C into C",
    "real part of a complex function" ~= "real-valued component u(x,y)",
    "imaginary part of a complex function" ~= "real-valued component v(x,y)",
    "holomorphic complex function" ~= "complex function differentiable throughout an open domain",
    "entire function" ~= "complex function holomorphic on all of C"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    function_domain_and_codomain_must_be_preserved: true,
    complex_limit_requires_all_approach_paths_to_agree: true,
    complex_differentiability_is_stronger_than componentwise_real_differentiability: true,
    holomorphicity_requires_open_domain: true,
    complex_logarithm_requires_branch_choice_for_single_valued_form: true,
    argument_is_undefined_at_zero: true,
    quotient_is_holomorphic_only_where_denominator_nonzero: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    component_decomposition: 0.99,
    limit_continuity_relation: 0.99,
    derivative_holomorphy_relation: 0.99,
    cauchy_riemann_relation: 0.99,
    branch_function_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(complex_function)
    -> identify(domain_and_codomain)
    -> decompose_into(real_and_imaginary_components_if_useful)
    -> identify(elementary_polynomial_rational_or_branch_structure)
    -> analyse(limit_and_continuity_if_requested)
    -> test(complex_differentiability)
    -> apply(cauchy_riemann_with_regularities_if_relevant)
    -> classify(holomorphic_analytic_entire_or_meromorphic)
    -> analyse(zeros_singularities_or_inverse_structure_if_requested)
    -> preserve(branch_and_domain_information)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed complex-function semantic graph",
    primary_entity: COMPLEX_FUNCTION,
    preserve: [
        domain,
        codomain,
        function_formula,
        real_component,
        imaginary_component,
        branch_choice,
        zeros,
        singularities
    ],
    infer_through: [
        complex_limits,
        continuity,
        complex_derivatives,
        cauchy_riemann_equations,
        holomorphic_functions,
        analytic_functions,
        elementary_complex_functions,
        conformal_maps
    ],
    avoid: [
        ignoring_function_domain,
        assuming_path_specific_limit_is complex limit,
        treating_real_differentiability_as holomorphy,
        using_log_or_power_without branch information,
        evaluating_argument_at zero,
        ignoring_denominator_zeros_in rational functions
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + COMPLEX_FUNCTION
    + DOMAIN
    + REAL_PART
    + IMAGINARY_PART
    + COMPLEX_LIMIT
    + COMPLEX_DERIVATIVE
    + CAUCHY_RIEMANN_EQUATIONS
    + HOLOMORPHIC_FUNCTION
    + ANALYTIC_FUNCTION
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> COMPLEX_FUNCTION
    -> FUNCTION_IDENTIFICATION
    -> DOMAIN_CODOMAIN_ANALYSIS
    -> COMPONENT_DECOMPOSITION
    -> FUNCTION_TYPE_CLASSIFICATION
    -> LIMIT_CONTINUITY_ANALYSIS
    -> COMPLEX_DIFFERENTIABILITY_ANALYSIS
    -> CAUCHY_RIEMANN_ANALYSIS
    -> HOLOMORPHY_CLASSIFICATION
    -> ZERO_SINGULARITY_INVERSE_ANALYSIS
    -> BRANCH_DOMAIN_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW