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 sum of a complex power series is analytic throughout its open disc of convergence
Statement
The sum of a complex power series is analytic on its open disc of convergence.
Facts & Assumptions
Given: A complex power-series sum on its open disc .
At every interior point , the sum re-expands as a convergent power series on a positive-radius disc about (A complex power-series sum re-expands about every interior point, at least to the distance from that point to the original boundary).
Analyticity means local representation by a convergent complex power series (Complex analytic functions as locally representable by convergent power series).
Proof
Let . Its distance to the original boundary is positive, and [L1] supplies a power-series representation of on a disc about .
This is exactly analyticity at by [L2]. Since was arbitrary, is analytic on , including the entire-radius case.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 22 results over 8 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
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)