Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02
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 from the complex exponential

Definition

For z∈Cz\in\mathbb C, define sin⁡z=exp⁡(iz)−exp⁡(−iz)2i,cos⁡z=exp⁡(iz)+exp⁡(−iz)2,\sin z=\frac{\exp(iz)-\exp(-iz)}{2i},\quad \cos z=\frac{\exp(iz)+\exp(-iz)}2, sinh⁡z=exp⁡z−exp⁡(−z)2,cosh⁡z=exp⁡z+exp⁡(−z)2.\sinh z=\frac{\exp z-\exp(-z)}2,\quad \cosh z=\frac{\exp z+\exp(-z)}2. The conventions and prerequisite facts used below are recorded in The complex exponential by its power series, The complex numbers form a field, and every nonzero x+iyx+iy has inverse (x−iy)/(x2+y2)(x-iy)/(x^2+y^2).

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 9 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