Definition:Left-Total Relation/Multifunction/Branch/Principal Branch

Definition

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 $f$.

Also see

• Definition:Principal Range
• Definition:Principal Argument

Notation

For some standard multifunctions it is conventional to distinguish the principal branch version by denoting it with a capital letter, for example:

$\Ln$

for the principal branch of the complex logarithm function $\ln$.

Linguistic Note

The word principal is (except in the context of economics) an adjective which means main.

Do not confuse with the word principle, which is a noun.

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