05 - Multivalued Functions

Multi-Valued Function / Multifunction

Multi-valued function or Multifunction on a subset $U\subseteq \mathbb{C}$ is map

$$f:U\to \mathcal{P}(\mathbb{C})$$

assigning to each point in $U$ a subset of the complex numbers (refer to Power set)

Branch

Let $f:U\to \mathcal{P}(\mathbb{C})$ be a multifunction

Branch of $f$ on subset $V\subseteq U$ is a function

$$g:V\to \mathbb{C}\text{ such that }g(z)\in f(z)\text{ for all }z\in V$$

If $g$ is continuous on $V$ then it is a continuous branch of $f$
If $g$ is holomorphic on $V$ then it is a holomorphic branch of $f$

Branch Point

Suppose that $f:U\to \mathcal{P}(\mathbb{C})$ is a multi-valued function defined on an open subset $U$ of $\mathbb{C}$

$z_0\in U$ is not a Branch Point of $f$ if there is an open disk $D\subseteq U$ containing $z_0$ such that
There exists a holomorphic branch of $f$ defined on $D\setminus \{z_0\}$

Otherwise $z_0$ is called a Branch Point

When $\mathbb{C}\setminus U$ is bounded then $f$ does not have a branch point at $\infty$ if
There is a holomorphic branch of $f$ defined on $\mathbb{C}\setminus B(0,R)\subseteq U$ for some $r>0$
Otherwise $\infty$ is a branch point of $f$

Notation for Multi-functions

Use square brackets to denote multifunction

$z^{1/2}$ Consider $z\mapsto z^{1/2}$ as a multifunction then write

$$[z^{1/2}]=\{w\in \mathbb{C}:w^2=z\}$$