TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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A continuous function holomorphic away from one point on a star-shaped domain has a primitive and zero closed-contour integrals

Statement

Let $U \subseteq C$ be open and star-shaped, let $p \in U$ , and let $f : U \to C$ be continuous and holomorphic on $U ∖ {p}$ . Then $f$ has a primitive on $U$ , and for every closed rectifiable contour $γ$ in $U$ ,

$\int_{γ} f (z) d z = 0.$

Facts & Assumptions

Given: An open star-shaped set $U$ , a point $p \in U$ , and a continuous function $f : U \to C$ holomorphic away from $p$ .

[L1]

Under these hypotheses, the integral of $f$ around every filled triangle contained in $U$ is zero (Goursat's triangle theorem remains valid for a continuous function holomorphic away from one point).

[L2]

Vanishing triangle integrals construct a 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).

[L3]

A primitive $F$ of $f$ is holomorphic and has $F^{'} = f$ (A primitive of a complex function on an open set).

[L4]

If $F$ is holomorphic, $F^{'}$ is continuous, and $γ$ is closed and rectifiable, then the integral of $F^{'}$ around $γ$ is zero (The integral of a continuous complex derivative over every closed rectifiable contour is zero).

Proof

technique · direct

1.1L1

By [L1], every contained triangle integral vanishes.

2.1givenstep 1.1L2L3

Since $f$ is continuous and $U$ is star-shaped, [L2] gives a primitive $F$ , and [L3] makes explicit that $F$ is holomorphic and $F^{'} = f$ .

3.1givenstep 2.1L4∎

The derivative $F^{'} = f$ is continuous by hypothesis; for any given closed rectifiable contour $γ$ in $U$ , all hypotheses of [L4] are therefore satisfied, and $\int_{γ} f = 0$ .

Depends on

Used by

Cauchy's integral formula on a circle compactly contained in a disc of holomorphy Theorem

Dependency tree · two levels

15 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

E. Stein and R. Shakarchi, Complex Analysis, Ch. 2, Theorem 1.4 (standard reference, not scraped)