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 $\ln$.