Definition:Multifunction/Principal Branch
Definition
Let $A$ and $B$ be sets.
Let $f: A \to B$ be a multifunction on $A$.
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 $f$.
Principal Value
Let $x \in A$ be an element of the domain of $f$.
The principal value of $x$ is the element $y$ of the principal branch of $f$ such that $\map f x = y$.
Also see
Notation
For some standard multifunctions, it is conventional to distinguish the principal branch 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