Complex Analysis Ontology

flow "ComplexAnalysisOntology" v1.0

# ============================================================
# COMPLEX ANALYSIS ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
COMPLEX_ANALYSIS := {
    type: MathematicalDiscipline,
    domain: Mathematics,
    parent: MathematicalAnalysis,
    purpose: "Formal study of complex-valued functions of complex variables, including complex differentiability, holomorphic functions, contour integration, analytic continuation, singularities, residues and conformal geometry."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    COMPLEX_NUMBER,
    COMPLEX_PLANE,
    COMPLEX_FUNCTION,
    COMPLEX_LIMIT,
    COMPLEX_CONTINUITY,
    COMPLEX_DERIVATIVE,
    HOLOMORPHIC_FUNCTION,
    ANALYTIC_FUNCTION,
    CAUCHY_RIEMANN_EQUATIONS,
    CONTOUR_INTEGRAL,
    CAUCHY_INTEGRAL_THEOREM,
    CAUCHY_INTEGRAL_FORMULA,
    LAURENT_SERIES,
    SINGULARITY,
    RESIDUE,
    RESIDUE_THEOREM,
    CONFORMAL_MAPPING
}

# ---------- ONTOLOGICAL STRUCTURE ----------
COMPLEX_ANALYSIS
    -> branch_of(MATHEMATICAL_ANALYSIS)
    -> studies(COMPLEX_FUNCTIONS)
    -> extends(REAL_ANALYSIS)
    -> uses(TOPOLOGY_OF_COMPLEX_PLANE)
    -> uses(POWER_SERIES)
    -> uses(CONTOUR_INTEGRATION)
    -> supports(SPECTRAL_AND_PHYSICAL_APPLICATIONS)

# ---------- COMPLEX NUMBER ----------
z
    := x + i*y

x
    -> REAL_PART_OF(z)

y
    -> IMAGINARY_PART_OF(z)

i^2
    = -1

z
    -> element_of(C)

# ---------- COMPLEX CONJUGATE ----------
conjugate(z)
    := x - i*y

z * conjugate(z)
    =
|z|^2

CONJUGATION
    -> reflects(COMPLEX_PLANE_ACROSS_REAL_AXIS)

# ---------- MODULUS ----------
|z|
    :=
sqrt(
    x^2+y^2
)

MODULUS
    -> defines(METRIC)

d(z,w)
    := |z-w|

C
    -> COMPLETE_METRIC_SPACE

# ---------- ARGUMENT ----------
for
    z != 0

ARGUMENT(z)
    := theta
       modulo(2*pi)

z
    =
|z|*e^(i*theta)

ARGUMENT
    -> MULTIVALUED_MODULO_2PI

PRINCIPAL_ARGUMENT
    -> requires(CHOSEN_BRANCH_RANGE)

# ---------- COMPLEX FUNCTION ----------
f
    := mapping(
        D subset_of C
        -> C
    )

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

u,v
    -> REAL_VALUED_COMPONENT_FUNCTIONS

# ---------- 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_be_independent_of(PATH_OF_APPROACH)

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

iff

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

COMPLEX_CONTINUITY
    -> metric_continuity_in(C)

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

if
    limit_exists_for_all_complex_directions

COMPLEX_DIFFERENTIABILITY
    -> stronger_than(REAL_DIFFERENTIABILITY_OF_COMPONENT_MAP)

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

if

D open
AND
f complex_differentiable_at_every_point_of(D)

HOLOMORPHIC_FUNCTION
    -> core_entity_of(COMPLEX_ANALYSIS)

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

if

f holomorphic_on(C)

ENTIRE_FUNCTION
    -> globally_holomorphic_complex_function

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

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

if

f
    -> represented_by(CONVERGENT_POWER_SERIES)
       in(neighbourhood_of(z0))

in complex analysis:

HOLOMORPHIC
    <-> ANALYTIC

ANALYTICITY
    -> consequence_of(COMPLEX_DIFFERENTIABILITY_ON_OPEN_SET)

# ---------- CAUCHY-RIEMANN ----------
for
    f=u+iv

CAUCHY_RIEMANN_EQUATIONS:

u_x
    = v_y

u_y
    = -v_x

if
    f complex_differentiable
then

CR_EQUATIONS
    -> hold_at_point
       under(component_differentiability)

if
    first_partials_continuous_near_point
    AND CR_EQUATIONS_hold
then

f
    -> complex_differentiable_locally

# ---------- HARMONIC FUNCTIONS ----------
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
       when(CR_RELATION_HOLDS)

# ---------- CONTOUR ----------
CONTOUR
    := gamma:[a,b] -> C

gamma
    -> piecewise_smooth

CLOSED_CONTOUR
    -> gamma(a)=gamma(b)

SIMPLE_CONTOUR
    -> no_self_intersections_except_possible_endpoint_closure

# ---------- CONTOUR INTEGRAL ----------
integral_gamma(
    f(z) dz
)
    :=
integral_from_a_to_b(
    f(gamma(t))
    * gamma_prime(t)
    dt
)

ORIENTATION
    -> affects(SIGN_OF_INTEGRAL)

reversing_orientation
    -> negates(CONTOUR_INTEGRAL)

# ---------- PRIMITIVE ----------
F
    -> primitive_of(f)

if

F_prime
    = f

if
    f has_primitive_on_domain
then

integral_from_z0_to_z1(
    f(z) dz
)
    -> path_independent

and

closed_contour_integral
    = 0

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

and
    C closed_piecewise_smooth_contour_in_D

then

closed_integral_C(
    f(z) dz
)
    = 0

TOPOLOGY
    -> essential_to(GLOBAL_CONTOUR_RESULT)

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

and
    z0 inside positively_oriented_simple_closed_C

then

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

BOUNDARY_VALUES
    -> determine(INTERIOR_HOLOMORPHIC_VALUES)

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

HOLOMORPHIC
    -> INFINITELY_DIFFERENTIABLE

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

inside radius R:

POWER_SERIES
    -> converges_absolutely
    -> converges_uniformly_on_compact_subdisks
    -> differentiable_termwise
    -> integrable_termwise

# ---------- TAYLOR SERIES ----------
for holomorphic f:

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

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

within
    disk_to_nearest_singularity_or_boundary_of_analyticity

# ---------- ANALYTIC CONTINUATION ----------
ANALYTIC_CONTINUATION
    -> extends(HOLOMORPHIC_FUNCTION_TO_LARGER_DOMAIN)

IDENTITY_THEOREM
    -> ensures(UNIQUENESS_OF_CONTINUATION)
       on(CONNECTED_OVERLAPPING_DOMAINS)

# ---------- ISOLATED SINGULARITY ----------
z0
    -> ISOLATED_SINGULARITY

if

f holomorphic_on(
    0 < |z-z0| < r
)

but
    not_holomorphic_at(z0)

ISOLATED_SINGULARITY
    -> classified_as(
        REMOVABLE,
        POLE,
        ESSENTIAL
    )

# ---------- LAURENT SERIES ----------
f(z)
    =
sum_from_n_minus_infinity_to_infinity(
    a_n*(z-z0)^n
)

on
    annulus

PRINCIPAL_PART
    := negative_power_terms

LAURENT_SERIES
    -> classifies(ISOLATED_SINGULARITIES)

# ---------- RESIDUE ----------
RESIDUE(f,z0)
    := a_(-1)

where
    a_(-1)
        -> Laurent_coefficient_of((z-z0)^(-1))

for simple pole:

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

# ---------- RESIDUE THEOREM ----------
if
    f meromorphic
    inside_and_on(C)
    and no_singularity_on(C)

then

closed_integral_C(
    f(z) dz
)
    =
2*pi*i
*
sum(
    residues_inside_C
)

CONTOUR_INTEGRAL
    -> reducible_to(LOCAL_SINGULARITY_DATA)

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

|f|
    -> no_local_maximum_in_interior

MAXIMUM_MODULUS_PRINCIPLE
    -> rigidity_property

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

f
    -> CONSTANT

LIOUVILLE_THEOREM
    -> supports(PROOF_OF_FUNDAMENTAL_THEOREM_OF_ALGEBRA)

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

f
    -> OPEN_MAP

OPEN_MAPPING_THEOREM
    -> strong_geometric_property_of(HOLOMORPHIC_FUNCTIONS)

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

f
    -> CONFORMAL_AT(z0)

CONFORMAL_MAP
    -> preserves(ANGLES_AND_ORIENTATION_LOCALLY)

# ---------- BRANCHES ----------
MULTIVALUED_FUNCTIONS := {
    LOGARITHM,
    COMPLEX_POWER,
    ROOT_FUNCTIONS
}

BRANCH
    -> single_valued_choice_on(RESTRICTED_DOMAIN)

BRANCH_CUT
    -> removes(TOPOLOGICAL_OBSTRUCTION)

COMPLEX_LOG
    -> requires(BRANCH_SELECTION_FOR_SINGLE_VALUED_HOLOMORPHIC_FORM)

# ---------- APPLICATIONS ----------
COMPLEX_ANALYSIS
    -> applied_to(REAL_INTEGRAL_EVALUATION)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(FLUID_DYNAMICS)
    -> applied_to(ELECTROMAGNETISM)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(SIGNAL_PROCESSING)
    -> applied_to(NUMBER_THEORY)

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

COMPLEX_DIFFERENTIABILITY_ON_OPEN_SET
    -> HOLOMORPHIC_FUNCTION

HOLOMORPHIC_FUNCTION
    -> ANALYTIC_FUNCTION

CAUCHY_RIEMANN_EQUATIONS
    -> characterise(HOLOMORPHY_WITH_REGULARITY)

HOLOMORPHY
    -> enables(CAUCHY_THEOREMS)

LAURENT_SERIES
    -> classifies(SINGULARITY)

RESIDUE
    -> determines(CONTOUR_INTEGRAL)

NONZERO_DERIVATIVE
    + HOLOMORPHY
    -> CONFORMALITY

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "complex analysis" ~= "rigorous analysis of complex-variable functions",
    "holomorphic function" ~= "function complex differentiable throughout an open domain",
    "analytic function" ~= "function locally represented by a convergent complex power series",
    "entire function" ~= "function holomorphic on the whole complex plane",
    "residue" ~= "negative-first-power Laurent coefficient at an isolated singularity",
    "conformal mapping" ~= "locally angle-preserving holomorphic map with nonzero derivative"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    complex_derivative_requires_same_limit_from_all_complex_directions: true,
    holomorphicity_requires_open_domain: true,
    cauchy_riemann_sufficiency_requires_regularities: true,
    contour_theorems_require_valid_domain_topology_and_orientation: true,
    residue_theorem_requires_no_singularity_on_contour: true,
    conformality_requires_nonzero_complex_derivative: true,
    multivalued_functions_require_branch_handling: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    ontology_structure: 0.99,
    derivative_holomorphy_relation: 0.99,
    cauchy_riemann_relation: 0.99,
    contour_theorem_relation: 0.99,
    singularity_residue_relation: 0.99,
    conformal_branch_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(complex_function_and_domain)
    -> identify(component_functions_if_needed)
    -> analyse(complex_limit_and_continuity)
    -> test(complex_differentiability)
    -> apply(cauchy_riemann_with_regularities_if_useful)
    -> classify(holomorphic_analytic_entire_or_meromorphic)
    -> identify(contour_orientation_and_domain_topology)
    -> apply(cauchy_integral_results_if_valid)
    -> classify(singularities_using_laurent_series)
    -> compute(residues_and_contour_integrals_if_requested)
    -> analyse(conformal_or_branch_structure)
    -> validate(all_domain_topology_and_regularities)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed complex-analysis ontology graph",
    primary_entity: COMPLEX_ANALYSIS,
    preserve: [
        complex_domain,
        function,
        component_functions,
        derivative_status,
        contour,
        orientation,
        singularities,
        branch_choice
    ],
    infer_through: [
        complex_limits,
        holomorphic_functions,
        analytic_functions,
        cauchy_riemann_equations,
        contour_integrals,
        power_series,
        laurent_series,
        residues,
        conformal_maps
    ],
    avoid: [
        treating_real_differentiability_as holomorphy,
        ignoring_open_domain_requirement,
        applying_cauchy_riemann_as sufficient without regularity,
        ignoring_contour_orientation_or topology,
        using_residue_theorem_with singularity_on contour,
        ignoring_branch_selection_for logarithms_and_roots
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + COMPLEX_ANALYSIS
    + COMPLEX_NUMBER
    + COMPLEX_DERIVATIVE
    + HOLOMORPHIC_FUNCTION
    + ANALYTIC_FUNCTION
    + CAUCHY_RIEMANN_EQUATIONS
    + CONTOUR_INTEGRAL
    + LAURENT_SERIES
    + RESIDUE
    + RESIDUE_THEOREM
    + CONFORMAL_MAPPING
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> COMPLEX_ANALYSIS
    -> FUNCTION_DOMAIN_IDENTIFICATION
    -> COMPONENT_FUNCTION_ANALYSIS
    -> LIMIT_CONTINUITY_ANALYSIS
    -> DIFFERENTIABILITY_ANALYSIS
    -> HOLOMORPHY_ANALYSIS
    -> CAUCHY_RIEMANN_ANALYSIS
    -> CONTOUR_TOPOLOGY_ANALYSIS
    -> CAUCHY_THEOREM_ANALYSIS
    -> SINGULARITY_LAURENT_ANALYSIS
    -> RESIDUE_ANALYSIS
    -> CONFORMAL_BRANCH_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW