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

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## Definition

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

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

Let $\sequence {S_i}_{i \in I}$ 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

## 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 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