Taylor Series Ontology

flow "TaylorSeriesOntology" v1.0

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

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

# ---------- CORE ENTITY ----------
TAYLOR_SERIES := {
    type: ComplexPowerSeriesRepresentation,
    domain: ComplexAnalysis,
    purpose: "Represent a holomorphic function locally as a power series whose coefficients are determined by the function's complex derivatives at an expansion centre."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    TAYLOR_SERIES,
    ANALYTIC_FUNCTION,
    HOLOMORPHIC_FUNCTION,
    POWER_SERIES,
    COMPLEX_DERIVATIVE,
    EXPANSION_CENTRE,
    TAYLOR_COEFFICIENT,
    RADIUS_OF_CONVERGENCE,
    CAUCHY_INTEGRAL_FORMULA,
    CAUCHY_ESTIMATE,
    SINGULARITY,
    ANALYTIC_CONTINUATION,
    MACLAURIN_SERIES,
    REMAINDER
}

# ---------- BASIC FORM ----------
let
    f
        -> holomorphic_near(z0)

then

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

where

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

TAYLOR_SERIES
    -> POWER_SERIES_ABOUT(z0)

# ---------- EXPANSION CENTRE ----------
z0
    -> EXPANSION_CENTRE

(z-z0)
    -> LOCAL_COMPLEX_DISPLACEMENT

TAYLOR_SERIES
    -> encodes(LOCAL_BEHAVIOUR_AROUND_CENTRE)

# ---------- COEFFICIENTS ----------
TAYLOR_COEFFICIENT
    := a_n

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

DERIVATIVES_AT_CENTRE
    -> uniquely_determine(TAYLOR_COEFFICIENTS)

# ---------- FIRST TERMS ----------
f(z)
    =
f(z0)
+
f_prime(z0)(z-z0)
+
f_double_prime(z0)/2! * (z-z0)^2
+
...

CONSTANT_TERM
    = f(z0)

LINEAR_TERM
    = f_prime(z0)(z-z0)

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

iff

f
    -> equals_its_Taylor_series_in_some_neighbourhood_of(z0)

in complex analysis:

HOLOMORPHIC
    <-> ANALYTIC

HOLOMORPHY
    -> guarantees(TAYLOR_REPRESENTATION_LOCALLY)

# ---------- MACLAURIN SERIES ----------
if
    z0 = 0

then

TAYLOR_SERIES
    -> MACLAURIN_SERIES

f(z)
    =
sum_from_n_0_to_infinity(
    f^(n)(0)/n!
    * z^n
)

MACLAURIN
    -> Taylor_series_centred_at_zero

# ---------- CAUCHY COEFFICIENT FORMULA ----------
if
    f holomorphic
    on neighbourhood_of_closed_disk
    centred_at(z0)

then

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

CAUCHY_INTEGRAL_FORMULA
    -> determines(TAYLOR_COEFFICIENTS)

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

TAYLOR_COEFFICIENT
    -> contour_integral_of(FUNCTION_VALUES)

# ---------- RADIUS OF CONVERGENCE ----------
TAYLOR_SERIES
    -> has(RADIUS_OF_CONVERGENCE_R)

inside:
    |z-z0| < R
        -> ABSOLUTE_CONVERGENCE

outside:
    |z-z0| > R
        -> DIVERGENCE

boundary:
    |z-z0| = R
        -> requires(SEPARATE_ANALYSIS)

# ---------- NEAREST SINGULARITY ----------
for analytic function represented by Taylor series:

R
    -> determined_by(distance_to_nearest_complex_singularity_or_analytic_obstruction)

SINGULARITY_LOCATION
    -> controls(MAXIMAL_DISK_OF_TAYLOR_CONVERGENCE)

REAL_FORMULA
    -> may_hide(NEARBY_COMPLEX_SINGULARITIES)

# ---------- CAUCHY-HADAMARD ----------
for power series coefficients a_n:

1/R
    =
limsup_(n->infinity)(
    |a_n|^(1/n)
)

CAUCHY_HADAMARD_FORMULA
    -> computes(RADIUS_FROM_COEFFICIENTS)

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

then

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

therefore

|a_n|
    <=
M/r^n

CAUCHY_ESTIMATES
    -> bound(TAYLOR_COEFFICIENTS)

# ---------- UNIFORM CONVERGENCE ----------
for every
    rho < R

TAYLOR_SERIES
    -> uniformly_convergent_on(
        |z-z0| <= rho
    )

COMPACT_SUBDISK
    -> supports(UNIFORM_CONVERGENCE)

# ---------- TERM-BY-TERM DIFFERENTIATION ----------
inside
    |z-z0| < R

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

DERIVATIVE_SERIES
    -> same_radius_of_convergence

# ---------- TERM-BY-TERM INTEGRATION ----------
inside convergence disk:

integral(
    sum(
        a_n*(z-z0)^n
    )
    dz
)
=
sum(
    a_n/(n+1)
    * (z-z0)^(n+1)
)

plus
    CONSTANT

INTEGRATED_SERIES
    -> same_radius_of_convergence

# ---------- TAYLOR POLYNOMIAL ----------
TAYLOR_POLYNOMIAL_N
    :=
sum_from_n_0_to_N(
    f^(n)(z0)/n!
    * (z-z0)^n
)

TAYLOR_POLYNOMIAL
    -> finite_approximation_to(f)

# ---------- REMAINDER ----------
REMAINDER_N(z)
    :=
f(z)
-
TAYLOR_POLYNOMIAL_N(z)

inside convergence disk:

REMAINDER_N(z)
    -> 0
       as(N -> infinity)

REMAINDER
    -> approximation_error

# ---------- CAUCHY REMAINDER CONTROL ----------
if
    f holomorphic_on_larger_disk
    AND bounded_by(M)
    on circle radius r

then

Taylor_tail
    -> bounded_using(GEOMETRIC_SERIES_AND_CAUCHY_ESTIMATES)

ERROR_BOUND
    -> depends_on(
        M,
        |z-z0|,
        r,
        N
    )

# ---------- EXPONENTIAL SERIES ----------
exp(z)
    =
sum_from_n_0_to_infinity(
    z^n/n!
)

R
    = infinity

exp(z)
    -> ENTIRE

# ---------- SINE SERIES ----------
sin(z)
    =
sum_from_n_0_to_infinity(
    (-1)^n
    * z^(2n+1)/(2n+1)!
)

R
    = infinity

sin(z)
    -> ENTIRE

# ---------- COSINE SERIES ----------
cos(z)
    =
sum_from_n_0_to_infinity(
    (-1)^n
    * z^(2n)/(2n)!
)

R
    = infinity

cos(z)
    -> ENTIRE

# ---------- GEOMETRIC SERIES ----------
1/(1-z)
    =
sum_from_n_0_to_infinity(
    z^n
)

for
    |z| < 1

nearest_singularity
    -> z=1

R
    = 1

# ---------- LOGARITHM SERIES ----------
Log(1+z)
    =
sum_from_n_1_to_infinity(
    (-1)^(n+1)
    * z^n/n
)

for
    |z| < 1
    under(CHOSEN_LOG_BRANCH_NEAR_1)

BRANCH_STRUCTURE
    -> must_be_preserved

# ---------- BINOMIAL SERIES ----------
(1+z)^alpha
    =
sum_from_n_0_to_infinity(
    binomial(alpha,n)
    * z^n
)

typically
    |z| < 1

with
    branch_choice_when(alpha_not_integer)

BINOMIAL_SERIES
    -> complex_power_series_representation

# ---------- ZERO MULTIPLICITY ----------
if
    f(z0)=0

and first nonzero Taylor coefficient
    is a_m

then

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

where

g(z0)
    = a_m != 0

m
    -> ORDER_OF_ZERO

TAYLOR_SERIES
    -> reveals(ZERO_MULTIPLICITY)

# ---------- IDENTITY THEOREM RELATION ----------
if
    all derivatives_of_f_at(z0)
        = 0

and
    f holomorphic_on_connected_domain

then

f
    -> IDENTICALLY_ZERO

TAYLOR_DATA_AT_ONE_POINT
    -> determines(HOLOMORPHIC_FUNCTION_LOCALLY)
    -> extends_by(IDENTITY_THEOREM)

# ---------- ANALYTIC CONTINUATION ----------
Taylor series about z0
    -> valid_on(MAXIMAL_DISK_TO_NEAREST_OBSTRUCTION)

choose new centre z1
inside disk

-> form(new_Taylor_series)

CHAIN_OF_OVERLAPPING_TAYLOR_DISKS
    -> supports(ANALYTIC_CONTINUATION)

# ---------- TAYLOR VS LAURENT ----------
TAYLOR_SERIES
    -> nonnegative_integer_powers_only

LAURENT_SERIES
    -> may_include(NEGATIVE_INTEGER_POWERS)

if
    function holomorphic_at_centre
then

LAURENT_PRINCIPAL_PART
    = 0

and

LAURENT_SERIES
    -> TAYLOR_SERIES

# ---------- APPLICATIONS ----------
TAYLOR_SERIES
    -> applied_to(FUNCTION_APPROXIMATION)
    -> applied_to(ANALYTIC_CONTINUATION)
    -> applied_to(SINGULARITY_LOCATION)
    -> applied_to(COMPLEX_DIFFERENTIATION)
    -> applied_to(CONTOUR_INTEGRATION)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(NUMERICAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
HOLOMORPHIC_FUNCTION
    -> has(LOCAL_TAYLOR_SERIES)

DERIVATIVES_AT_CENTRE
    -> determine(COEFFICIENTS)

COEFFICIENTS
    -> determine(POWER_SERIES)

NEAREST_SINGULARITY
    -> limits(RADIUS_OF_CONVERGENCE)

TAYLOR_SERIES
    -> differentiable_term_by_term

TAYLOR_SERIES
    -> integrable_term_by_term

FIRST_NONZERO_COEFFICIENT
    -> determines(ZERO_MULTIPLICITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Taylor series" ~= "power series whose coefficients are derivatives at the expansion centre divided by factorials",
    "Maclaurin series" ~= "Taylor series centred at zero",
    "Taylor coefficient" ~= "f^(n)(z0)/n!",
    "radius of convergence" ~= "largest disk radius on which the power series converges",
    "Taylor polynomial" ~= "finite truncation of a Taylor series",
    "order of zero" ~= "index of the first nonzero Taylor coefficient at a zero"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    complex_taylor_series_represents_function_when_function_is_holomorphic_near_centre: true,
    coefficients_are_uniquely_determined_by_derivatives: true,
    radius_of_convergence_is_limited_by_nearest_analytic_obstruction: true,
    boundary_points_require_separate_convergence_analysis: true,
    termwise_differentiation_and_integration_are_valid_inside_radius: true,
    branch_dependent_functions_require_consistent_branch_choice: true,
    taylor_series_contains_no_negative_powers: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    coefficient_relation: 0.99,
    holomorphic_relation: 0.99,
    radius_singularity_relation: 0.99,
    termwise_operation_relation: 0.99,
    zero_multiplicity_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_and_expansion_centre)
    -> verify(holomorphy_near_centre)
    -> compute(derivatives_or_equivalent_coefficients)
    -> construct(taylor_series)
    -> determine(radius_of_convergence)
    -> identify(nearest_singularity_or_obstruction)
    -> test(target_point_against_convergence_disk)
    -> truncate_and_estimate(remainder_if_requested)
    -> perform(termwise_differentiation_or_integration_if_requested)
    -> analyse(zero_multiplicity_or_continuation_if_relevant)
    -> preserve(branch_information)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Taylor-series semantic graph",
    primary_entity: TAYLOR_SERIES,
    preserve: [
        function,
        expansion_centre,
        coefficients,
        derivatives,
        radius_of_convergence,
        nearest_singularity,
        truncation_order,
        branch_choice
    ],
    infer_through: [
        analytic_functions,
        holomorphic_functions,
        cauchy_integral_formula,
        cauchy_estimates,
        power_series,
        zero_multiplicity,
        analytic_continuation,
        laurent_series
    ],
    avoid: [
        constructing_taylor_representation_without local holomorphy,
        assuming_series_valid_beyond radius,
        ignoring_complex_singularities_when radius estimated from real formula,
        performing_termwise_operations_on boundary_without justification,
        confusing_taylor_with laurent series,
        ignoring_branch_choice_for logarithms_and_noninteger powers
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + TAYLOR_SERIES
    + ANALYTIC_FUNCTION
    + TAYLOR_COEFFICIENT
    + RADIUS_OF_CONVERGENCE
    + CAUCHY_INTEGRAL_FORMULA
    + CAUCHY_ESTIMATE
    + TAYLOR_POLYNOMIAL_N
    + REMAINDER
    + ANALYTIC_CONTINUATION
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> TAYLOR_SERIES
    -> FUNCTION_CENTRE_IDENTIFICATION
    -> HOLOMORPHY_VALIDATION
    -> COEFFICIENT_ANALYSIS
    -> SERIES_CONSTRUCTION
    -> RADIUS_SINGULARITY_ANALYSIS
    -> TARGET_POINT_CONVERGENCE_ANALYSIS
    -> TRUNCATION_REMAINDER_ANALYSIS
    -> TERM_OPERATION_ANALYSIS
    -> ZERO_CONTINUATION_ANALYSIS
    -> BRANCH_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW