Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16
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  • 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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Every complex analytic function is holomorphic

Statement

Every function analytic on an open set U⊆C is holomorphic on U.

Facts & Assumptions

Given: A function f analytic on an open set U.

[L1]

Analyticity at a supplies a convergent power series representing f on a disc about a (Complex analytic functions as locally representable by convergent power series).

[L2]

A complex power-series sum is holomorphic throughout its open disc of convergence (Inside its disc of convergence a complex power series is holomorphic and may be differentiated term by term).

Proof

technique · direct
1.1L1

Let a∈U. By [L1], f agrees near a with a convergent power series.

2.1step 1.1L2

By [L2], that power-series sum is complex differentiable at a, hence so is f.

3.1step 2.1∎

Since a was arbitrary, f is holomorphic on U; if U is empty, this conclusion is vacuous.

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources