CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives

Statement

The functions $\sin, \cos, \sinh, \cosh$ are entire and satisfy $\sin^{'} = \cos, \cos^{'} = - \sin, \sinh^{'} = \cosh, \cosh^{'} = \sinh .$

Facts & Assumptions

Given: The four entire power series of The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series.

[L1]

A complex power series may be differentiated term by term inside its radius (Inside its disc of convergence a complex power series is holomorphic and may be differentiated term by term).

Proof

technique · direct

1.1

Apply [L1] to each of the four infinite-radius series and cancel the positive integer factor against the factorial.

L1algebra

2.1

Shifting the resulting indices gives respectively the series for $\cos, - \sin, \cosh, \sinh$ . Infinite radius makes each function entire.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Stein and R. Shakarchi, Complex Analysis, Ch. 1 (standard reference, not scraped)