Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16
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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
1.1

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

L1
2.1

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

step 1.1L2
3.1

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

step 2.1∎

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: 19 results over 6 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