Alphabeta MathHow 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 complex exponential series converges absolutely for every complex argument
Statement
For every , the series converges absolutely. The conventions and prerequisite facts used below are recorded in The complex exponential by its power series, The exponential series converges absolutely for every real argument, Every absolutely convergent complex series converges, and rearrangements preserve its sum.
Facts & Assumptions
Given: .
Proof
Its modulus series is , the real exponential series at the nonnegative real .
The real infinite-radius lemma and complex absolute-convergence theorem give the result.
Depends on
Used by
- Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ Theorem
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential Theorem
Cited to discharge well-definedness by The complex exponential by its power series.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 74 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
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)