Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Complex de Moivre formula for every integer exponent

Statement

Let j:Z→Qj:\mathbb Z\to\mathbb Q be the integer embedding of The integers embed in the rationals, let ιR:Q→R\iota_{\mathbb R}:\mathbb Q\to\mathbb R be the ordered-field embedding of The unique embedding of ℚ into an ordered field, and put κR:=ιR∘j\kappa_{\mathbb R}:=\iota_{\mathbb R}\circ j. For every integer mm and real θ\theta, (cos⁡θ+isin⁡θ)m=cos⁡(κR(m)θ)+isin⁡(κR(m)θ).(\cos\theta+i\sin\theta)^m=\cos(\kappa_{\mathbb R}(m)\theta)+i\sin(\kappa_{\mathbb R}(m)\theta). The conventions and prerequisite facts used below are recorded in 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, The complex numbers form a field, and every nonzero x+iyx+iy has inverse (x−iy)/(x2+y2)(x-iy)/(x^2+y^2), Integer powers in the complex field.

Facts & Assumptions

Given: An integer mm and real θ\theta.

Proof

technique · direct
1.1

Euler's formula identifies the base with exp⁡(iθ)\exp(i\theta).

given
1.2

Repeated addition handles nonnegative powers by the exponential addition law; inverses handle negative powers.

given
2.1

Euler's formula at κR(m)θ\kappa_{\mathbb R}(m)\theta gives the displayed result.

given∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 118 results over 31 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