Complex Analysis MT23, Multifunctions and branch cuts
Flashcards
Can you define a multifunction on $U \subseteq \mathbb C$?
A map $f : U \to \mathcal P(\mathbb C)$.
Suppose $f : U \to \mathcal P(\mathbb C)$ is a multifunction. Then what is a branch $g : V \to \mathbb C$ where $V \subseteq U$?
Any function $g$ such that for all $z \in V$, $g(z) \in f(z)$.
How many solutions does $z^\alpha = w$ where $\alpha \in \mathbb Q$ and $z, w \in \mathbb C$ have?
$n$ inverses.
How many solutions does $z^\alpha = w$ where $\alpha \in \mathbb Q^C$ and $z, w \in \mathbb C$ have?
Infinitely many inverses.
Suppose:
- $f : U \to \mathcal P(\mathbb C)$ is a multi-valued function
- $U$ is an open subset of $\mathbb C$
What does it mean for a point $z _ 0$ to be a branch point of $f$?
$z _ 0$ is not a branch point if there is an open disc $D \subseteq U$ containing $z _ 0$ such that there is a holomorphic branch of $f$ defined on $D \setminus \{z _ 0\}$ .
$z _ 0$ is a branch point otherwise.