TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Every holomorphic function on a star-shaped domain has a primitive

Statement

Let $U \subseteq C$ be open and star-shaped with respect to $a \in U$ . Every holomorphic function $f : U \to C$ has the primitive

$F (z) = \int_{ℓ_{a z}} f (ζ) d ζ .$

Facts & Assumptions

Given: An open set $U$ star-shaped with respect to $a \in U$ , and a holomorphic $f : U \to C$ .

[L1]

A holomorphic function is continuous (Complex differentiability at a point implies continuity there).

[L2]

Goursat's theorem makes the integral of a holomorphic function around every filled triangle in its domain zero (Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain).

[L3]

Vanishing triangle integrals construct the displayed primitive for a continuous function on a star-shaped domain (Vanishing integrals around triangles construct a primitive for a continuous function on a star-shaped domain).

Proof

technique · direct

1.1

By [L1], $f$ is continuous, and by [L2] its integral around every filled triangle contained in $U$ is zero.

L1L2

2.1

The hypotheses of [L3] now hold, so the displayed function is holomorphic on $U$ and has derivative $f$ .

step 1.1L3∎

Depends on

Used by

Cauchy's theorem on a star-shaped domain: every closed rectifiable contour integral of a holomorphic function is zero Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 13 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

E. Stein and R. Shakarchi, Complex Analysis, Ch. 2, Theorem 2.1 (standard reference, not scraped)