TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 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.

The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over $C$

Statement

For every $z \in C$ , $\cos z = \cosh (i z), \sin z = - i \sinh (i z), \cosh z = \cos (i z), \sinh z = - i \sin (i z) .$ The complex functions restrict to their real sine, cosine, hyperbolic sine and hyperbolic cosine series on the real axis. The conventions and prerequisite facts used below are recorded in Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential, Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential.

Facts & Assumptions

Given: A complex number $z$ .

Proof

technique · direct

1.1

Substitute $i z$ and $- i z$ into the four definitions and simplify $i^{2} = - 1$ .

algebra

2.1

On real arguments, group the exponential series into even and odd terms as in Euler's formula.

given∎

Depends on

Used by

1/sin(1/z) has a nonisolated singularity at 0 Counterexample
Agreement accumulating only at the boundary does not force a holomorphic identity Counterexample
sin(1/z) has an essential singularity at 0 Counterexample
Complex sine is unbounded on the imaginary axis Example
FALSE: an entire function bounded on the real axis is constant False statement
Complex sine and cosine are unbounded on the complex plane Theorem
The complex Pythagorean identity by the identity theorem Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)

The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over CC