Definition:Left-Total Relation/Multifunction/Branch

Definition

Let $D \subseteq \C$ be a subset of the complex numbers.

Let $f: D \to \C$ be a multifunction on $D$.

Let $\family {S_i}_{i \mathop \in I}$ be a partitioning of the codomain of $f$ such that:

$\forall i \in I: f \restriction_{D \times S_i}$ is a mapping.

Then each $f \restriction_{D \times S_i}$ is a branch of $f$.

Principal Branch

Let $D \subseteq \C$ be a subset of the complex numbers.

Let $f: D \to \C$ be a multifunction on $D$.

Let $\sequence {S_i}_{i \mathop \in I}$ be a partitioning of the codomain of $f$ into branches.

It is usual to distinguish one such branch of $f$ from the others, and label it the principal branch of $f$.

Branch Point

A branch point of $f$ is a point $a$ in $U$ such that:

$f$ has more than one value at one or more points in every neighborhood of $a$

$f$ has exactly one value at $a$ itself.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: Single- and Multiple-Valued Functions