KU Math 800: Lecture 2/11/2026
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Homotopy and Simply-Connectedness
[definition=%counter%] Let $\gamma_0,\gamma_1:[a, b] \to \CC$ are two curves contained in an open set $\Omega \subseteq \CC$ such that $\gamma_0$ and $\gamma_1$ have the same endpoints. Then $\gamma_0$ and $\gamma_1$ are homotopic in $\Omega$ if there exists $\gamma_s:[a, b] \to \CC$, for $0 \le s \le 1$, such that
\[\eval{\gamma_S}_{s=0} = \gamma_0 \quad\text{ and }\quad \eval{\gamma_S}_{s=1} = \gamma_1.\][/definition]
[theorem=%counter% (Theorem 5.1)] If $f$ is holomorphic in $\Omega$, then
\[\int_{\gamma_0} f(z) \dd{z} = \int_{\gamma_1} f(z) \dd{z}\]where $\gamma_0$ and $\gamma_1$ are homotopic inside $\Omega$. [/theorem]
[definition=%counter%] An open set $\Omega \subseteq \CC$ is simply-connected if it has trivial fundamental group. [/definition]
[theorem=%counter% (Theorem 5.2)] Any holomorphic function in a simply-connected domain $\Omega$ has a primitive. [/theorem]
[proof] Fix $z_0 \in \Omega$. Then define
\[F(z) = \int_{\gamma} f(w) \dd{w}\]where $\gamma$ is any curve joining $z_0 \to z$. By the previous theorem, this is well-defined. Then
\[F(z + h) = \int_{\eta} f(w) \dd{w}\]where $\eta$ is a line segment $z_0 \to z + h$. Then we argue similarly in the case of a disc to show that
\(\lim_{h\to 0} \frac{F(z + h) - F(z)}{h} = f(z).\) [/proof]
[corollary=%counter%] If $f$ is holomorphic in $\Omega$, a simply-connected domain, then
\[\int_{\gamma} f(z) \dd{z} = 0\]for any closed curved $\gamma \subseteq \Omega$. [/corollary]
Complex Logarithm
[theorem=%counter%] Suppose $\Omega$ is simply-connected with $1 \in \Omega$ and $0 \notin \Omega$. Then there exists a function $F$ such that
- $F$ is holomorphic in $\Omega$
- $e^{F(z)} = z$ for all $z \in \Omega$
- $F(r) = r$ if $r \in \RR$ and close to $1$. [/theorem]
[proof] Let $F$ be the primitive of $\frac{1}{z}$ in $\Omega$ and use the Theorem above: Let
\[F(z) = \int_\gamma \frac{1}{w} \dd{w}\]with $\gamma$ a path $1 \to z$.
For (2), we just note that
\[\dv{z} (z e^{-F(z)}) = e^{-F(z)} - zF'(z) e^{-F(z)} = 0\]since $F’ = 1/z$. So $z e^{-F(z)}$ is a constant with $F(1) = 0$ and the claim follows.
For (3), we let $\gamma$ be the line segment from $1$ to $r$ in $\RR$ and the claim follows by usual calculus. [/proof]
[definition=%counter%] Such a choice of $F$ is a called a branch of the logarithm. [/definition]
[definition=%counter%] Let $\Omega = \CC \setminus (-\infty, 0]$. Then the $F$ we obtain from the existence theorem above is the principal branch of the logarithm. In particular, we get
\[F(z) = \log r + i\theta\]where $z = re^{i\theta}$ and $\theta \in (-\pi, \pi)$. [/definition]
[definition=%counter%] Let $\Omega$ be simply connected with $1 \in \Omega$ and $0 \notin \Omega$. Then we define
\[z^\alpha = e^{\alpha \log z}\]where $\alpha \in \CC$. [/definition]