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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

1.1givenL1choose

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

2.1givenstep 1.1L2∎

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

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

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