KU Math 800: Lecture 1/28/2026
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[definition=%counter%] Let $f:\Omega \to \CC$ be holomorphic. We say that $F:\Omega \to \CC$ is a primitive of $f$ if
\[F'(z) = f(z)\]for all $z \in \Omega$. [/definition]
[theorem=%counter% (1.3.2 in Stein-Shakarchi)] If $f:\Omega \to \CC$ has a primitive $F$ and $\gamma \subseteq \Omega$ starting at $a$ and ending at $b$, then
\[\int_{\gamma} f(z) \dd{z} = F(b) - F(a).\][/theorem]
[corollary=%counter% (1.3.3 in Stein-Shakarchi)] Furthermore, if $\gamma$ is closed, then $\int_\gamma f = 0$. [/corollary]
[theorem=%counter% (Cauchy’s Theorem for a Disk)] If $f$ is holomorphic in an open disk centered at $z_0$ with radius $r$, then
\[\int_{\gamma} f(z) \dd{z} = 0\]for every closed curved inside the disk. [/theorem]
To prove Cauchy’s Theorem for a disk, it suffices to show that a holomorphic function in an open disk admits a primitive. In single variable calculus, this is easy by the fundamental theorem. We can extend this result but we have to be a little careful. We omit the proof.
[corollary=%counter% (2.2.3 in Stein-Shakarchi)] If $f$ is holomorphic in the open set $\Omega \subseteq \CC$ containing a circle $C$ and its interior, then
\[\int_C f(z) \dd{z} = 0.\][/corollary]
[definition=%counter%] A toy contour is any closed curve where the notion of interior is obvious. In particular, we mean keyholes, semicircles, indented circles, sectors, parallelograms, etc. [/definition]
[theorem=%counter% (Cauchy’s Theorem on Toy Contours)] Cauchy’s theorem is true if we replace “open disk centered at $z_0$ with radius $r$” with a region enclosed by a toy contour. [/theorem]
[example=%counter% (Example 2.2.1 in Stein-Shakarchi)]
\[\int_0^\infty \frac{1 - \cos x}{x^2} \dd{x} = \frac{\pi}{2}.\][/example]
[proof] Let $f(z) = \frac{1 - e^{iz}}{z^2}$ and then use an indented contour then apply the usual Cauchy theorem and limit argument. [/proof]