Definition:Multifunction/Principal Branch

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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


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


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.