Skip to content

Generators, coefficients, and Fekete–Szegő functionals

Exact catalog lookup, generator-series expansion, Taylor coefficients, and the low-order Fekete–Szegő functional.

geometric_function_atlas.catalog

Curated Ma–Minda generators used by the published radius portfolio.

The built-in catalog mirrors the named and parametric generator classes of the website source catalog (gft.maminda at source commit :data:~geometric_function_atlas.version.SOURCE_ARTIFACT_COMMIT). Each entry is an exact SymPy expression in :data:~geometric_function_atlas.models.Z with bibliographic provenance; grid classes carry the same keys the website uses so an expansion can be cross-checked against the site.

list_generators() -> tuple[Generator, ...]

Return the built-in generators in stable key order.

get_generator(key: str) -> Generator

Return a built-in generator or fail with a discoverable error.

generator_artifact_version(generator: Generator) -> str

Return the catalog version or identify a caller-supplied definition.

generator_identity(generator: Generator) -> str

Return a collision-safe identity for a catalog or caller generator.

Built-ins retain their stable public key. Caller definitions include the full SHA-256 digest of the exact SymPy formula, so a reused catalog key cannot disguise a different mathematical object. Names and citations are provenance rather than part of the defining function.

geometric_function_atlas.coefficients

Exact coefficient operations for Ma–Minda generators.

MAX_TAYLOR_ORDER = 64 module-attribute

GeneratorSeriesResult dataclass

Exact generator coefficients with reproducibility metadata.

generator: Generator instance-attribute

order: int instance-attribute

coefficients: tuple[sp.Expr, ...] instance-attribute

method: str = 'exact_symbolic_taylor_series' class-attribute instance-attribute

evidence_status: str property

Return the package-derived computational evidence state.

verification_report: VerificationReport property

__init__(generator: Generator, order: int, coefficients: tuple[sp.Expr, ...], method: str = 'exact_symbolic_taylor_series') -> None

__post_init__() -> None

to_dict() -> dict[str, Any]

taylor_coefficients(generator: str | Generator, *, order: int) -> tuple[sp.Expr, ...]

Return exact B_1, …, B_order for phi(z) = 1 + Σ B_n z^n.

Custom :class:Generator objects are accepted so reproductions are not limited to the built-in catalog. The normalization phi(0) = 1 is checked exactly before coefficients are returned.

generator_series(generator: str | Generator, *, order: int) -> GeneratorSeriesResult

Return exact coefficients plus a structured provenance record.

geometric_function_atlas.fekete_szego

Exact Fekete–Szegő constants under Ma–Minda assumptions.

MAX_DECIMAL_PRECISION = 1000 module-attribute

MAX_RATIONAL_CHARACTERS = 128 module-attribute

MAX_RATIONAL_COMPONENT_BITS = 426 module-attribute

FeketeSzegoResult dataclass

Exact result plus the assumptions and provenance needed to review it.

generator: Generator instance-attribute

mu: sp.Rational instance-attribute

b1: sp.Expr instance-attribute

b2: sp.Expr instance-attribute

value: sp.Expr instance-attribute

method: str = 'ma_minda_fekete_szego_closed_form' class-attribute instance-attribute

evidence_status: str property

Return the package-derived computational evidence state.

verification_report: VerificationReport property

__init__(generator: Generator, mu: sp.Rational, b1: sp.Expr, b2: sp.Expr, value: sp.Expr, method: str = 'ma_minda_fekete_szego_closed_form') -> None

__post_init__() -> None

decimal(*, precision: int = 16) -> str

to_dict(*, precision: int = 16) -> dict[str, Any]

Return a JSON-serializable, evidence-typed record.

fekete_szego(generator: str | Generator, *, mu: int | str | sp.Rational) -> FeketeSzegoResult

Compute the exact sharp |a3 - mu*a2**2| constant.

For an admissible Ma–Minda generator phi(z) = 1 + B1*z + B2*z**2 + ..., the general formula is

B1/2 * max(1, |B2/B1 + (1 - 2*mu)*B1|).

The package checks the exact coefficient preconditions it can decide. Full analytic admissibility and literature novelty are deliberately outside this function's claim boundary.