Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck 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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exp⁡(x+iy)=ex(cos⁡y+isin⁡y), ∣exp⁡(x+iy)∣=ex, and eiπ+1=0

Statement

For real x,y, exp⁡(x+iy)=ex(cos⁡y+isin⁡y) and ∣exp⁡(x+iy)∣=ex. In particular eiπ+1=0. The conventions and prerequisite facts used below are recorded in exp⁡(z+w)=exp⁡z exp⁡w, and the complex exponential extends the real exponential, Euler's formula: exp⁡(iθ)=cos⁡θ+isin⁡θ for every real θ, Conjugation is an involutive real-field automorphism, zz‾=∣z∣2, and modulus is definite, multiplicative, and subadditive, Pythagorean and parity identities for all six trigonometric functions on their natural domains, The exponential is positive and satisfies exp⁡(−x)=1/exp⁡(x), Quarter-turn values and shifts by pi/2 and pi, Pi as twice the smallest positive zero of cosine.

Facts & Assumptions

Given: Reals x,y.

Proof

technique · direct
1.1given

Apply the addition law to x+iy and Euler's formula.

1.2given

Multiplicativity of modulus, the Pythagorean identity, and positivity of ex give the modulus formula.

2.1given∎

The defining quarter-turn value cos⁡π=−1 and sin⁡π=0 give Euler's identity.

Depends on

Dependency tree · 2 levels

Sources