Simply Connected Domains

Theorem (General Closed Curve): Suppose $f$ is analytic in a simply connected region $D$ and that $C$ is a smooth closed curve contained in $D$. Then

\[\int_C f = 0\]

Proof: TODO.

Analytic Function $\log z$

Definition: $f$ is an analytic branch of $\log z$ in a domain $D$ if

1. $f$ is analytic in $D$
2. $f$ is an inverse of the exponential function in $D$.

Theorem: Suppose $D$ is simply connected and $0 \not\in D$. Choose $z_0 \in D$ and fix a value of $\log z_0$. Then

\[f(z) = \int_{z_0}^z \frac{d\zeta}{\zeta} + \log z_0\]

is an analytic branch of $\log z$ in $D$.