CorollaryStatement: 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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Cauchy's theorem on a convex complex domain

Statement

Let $U$ be a complex domain whose image in $R^{2}$ is convex in the sense of Complex star-shaped and convex domains are the published Euclidean notions under the identification $C = R^{2}$ . If $f : U \to C$ is holomorphic and $γ$ is a closed rectifiable contour in $U$ , then

$\int_{γ} f (z) d z = 0.$

Facts & Assumptions

Given: A convex complex domain $U$ , a holomorphic $f : U \to C$ , and a closed rectifiable contour $γ$ in $U$ .

[L1]

A complex domain is nonempty and open, and every convex open subset of Euclidean space is star-shaped with respect to each of its points (A complex domain is a nonempty connected open subset of $C$ , Star-shaped open subsets of Euclidean space).

[L2]

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

Proof

technique · direct

1.1

By [L1], choose any $a \in U$ and regard $U$ as star-shaped with respect to $a$ .

givenL1choose

2.1

Now [L2] applied to the given $f$ and $γ$ gives the displayed zero integral.

givenstep 1.1L2∎

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: 36 results over 10 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

Tang-Kai Lee, Complex Analysis Notes, Section 2.1.2 (standard reference, not scraped)