# 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