KU Math 800: Lecture 1/26/2026
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[theorem=%counter% (Cauchy’s Theorem)] Let $\Omega \subseteq \CC$ be open and $\gamma \subseteq \Omega$ be a closed curve such that the interior of $\gamma$ is contained in $\Omega$. If $f:\Omega \to \CC$ is holomorphic, then
\[\int_\gamma f(z) \dd{z} = 0.\][/theorem]
[example=%counter%] The interior condition is strictly necessary. By interior, we mean the interior of the region enclosed by $\gamma$. Recall that
\[\int_{\partial \mathbb D} \frac{\dd{z}}{z} = 2\pi i.\][/example]
[theorem=%counter% (Goursat’ Theorem)] Let $\Omega \subseteq \CC$ and $\gamma \subseteq \Omega$ be a triangle whose interior is contained in $\Omega$. Then
\[\int_\gamma f(z) \dd{z} = 0.\][/theorem]