Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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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.

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

Statement

For every z∈Cz\in\mathbb C, cos⁡z=cosh⁡(iz),sin⁡z=−isinh⁡(iz),cosh⁡z=cos⁡(iz),sinh⁡z=−isin⁡(iz).\cos z=\cosh(iz),\qquad \sin z=-i\sinh(iz),\qquad \cosh z=\cos(iz),\qquad \sinh z=-i\sin(iz). 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⁡(iθ)=cos⁡θ+isin⁡θ\exp(i\theta)=\cos\theta+i\sin\theta for every real θ\theta, exp⁡(z+w)=exp⁡z exp⁡w\exp(z+w)=\exp z\,\exp w, and the complex exponential extends the real exponential.

Facts & Assumptions

Given: A complex number zz.

Proof

technique · direct
1.1

Substitute iziz and −iz-iz into the four definitions and simplify i2=−1i^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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 84 results over 15 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