ThmDex – An index of mathematical definitions, results, and conjectures.

Cauchy's theorem for a disc

Formulation 0

Let $D \subseteq \mathbb{C}$ be an D4995: Open complex disc such that

(i)	$f : D \to \mathbb{C}$ is a D1392: Holomorphic function on $D$
(ii)	$\gamma \subseteq C$ is an D5023: Oriented complex curve
(iii)	$\gamma$ is a D5646: Closed complex curve

Then \begin{equation} \int_{\gamma} f(z) \, d z = 0 \end{equation}

Proofs

Proof 0

Let $D \subseteq \mathbb{C}$ be an D4995: Open complex disc such that

(i)	$f : D \to \mathbb{C}$ is a D1392: Holomorphic function on $D$
(ii)	$\gamma \subseteq C$ is an D5023: Oriented complex curve
(iii)	$\gamma$ is a D5646: Closed complex curve

Result R3456: Holomorphic function on open disc has primitive on that disc shows that $f$ has a primitive $F : D \to \mathbb{C}$ on $D$. The claim is now a consequence of results

(i)	R1522: Holomorphic function is continuous
(ii)	R1564: Oriented complex curve integral of continuous function with a primitive on open set

$\square$