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

Statement

Let $U \subseteq C$ be open and star-shaped, let $f : U \to C$ be holomorphic, and let $γ$ be a closed rectifiable contour in $U$ . Then

$\int_{γ} f (z) d z = 0.$

Facts & Assumptions

Given: An open star-shaped set $U$ , a holomorphic $f : U \to C$ , and a closed rectifiable contour $γ$ in $U$ .

[L1]

Every holomorphic function on a star-shaped domain has a primitive (Every holomorphic function on a star-shaped domain has a primitive).

[L2]

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

[L3]

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

[L4]

If $F$ is holomorphic, $F^{'}$ is continuous, and $γ$ is closed and rectifiable, then $\int_{γ} F^{'} = 0$ (The integral of a continuous complex derivative over every closed rectifiable contour is zero).

Proof

technique · direct

1.1L1L2L3

By [L1] and [L2], there is a holomorphic $F$ on $U$ with $F^{'} = f$ ; by [L3], this derivative is continuous.

2.1givenstep 1.1L4∎

The given $γ$ is closed and rectifiable, so every hypothesis of [L4] holds and $\int_{γ} f = \int_{γ} F^{'} = 0$ .

Depends on

Used by

Dependency tree · two levels

12 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 2.2 (standard reference, not scraped)