TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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A complex function is holomorphic if and only if it is analytic

Statement

Let $U \subseteq C$ be open and let $f : U \to C$ . Then $f$ is holomorphic on $U$ if and only if it is analytic on $U$ in the local power-series sense of Complex analytic functions as locally representable by convergent power series.

Facts & Assumptions

Given: An open set $U \subseteq C$ and a function $f : U \to C$ .

[L1]

Every holomorphic function equals its Taylor series throughout the largest centred open disc contained in its domain (A holomorphic function equals its Taylor series throughout the largest centred disc in its domain).

[L2]

A function is analytic on an open set exactly when every point has a contained open disc on which the function equals a convergent complex power series centred at that point (Complex analytic functions as locally representable by convergent power series).

[L3]

Every function analytic on an open subset of $C$ is holomorphic there (Every complex analytic function is holomorphic).

Proof

technique · direct

1.1

For the holomorphic-to-analytic direction, if $f$ is holomorphic and $a \in U$ , [L1] gives a positive-radius disc about $a$ on which $f$ equals its Taylor series, so [L2] makes $f$ analytic at $a$ and hence on $U$ .

L1L2

1.2

For the analytic-to-holomorphic direction, the local power-series hypothesis of [L2] is exactly the hypothesis of [L3], which makes $f$ holomorphic on the same open set $U$ .

L2L3

2.1

Steps 1.1 and 1.2 prove both implications; when $U = \emptyset$ , both pointwise predicates hold vacuously, so the equivalence also includes the empty open set.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

B. V. Shabat, Introduction to Complex Analysis, Theorem 2.24 (standard reference, not scraped)
Matthias Weber, Complex Analysis, §2.2 (standard reference, not scraped)