Branch

A single-valued piece cut out of a multivalued function by restricting its output range.

A branch of a multivalued function is a single-valued function obtained by restricting the allowed outputs so that exactly one value is selected at each input in some domain -- turning the multivalued rule into an ordinary, single-valued one there. Different branches of the same multivalued function agree in shape but differ by which one of the infinitely many possible values they report.

Examples

Restricting $\arg z$ to values in $(-\pi,\pi]$ picks out one branch of the argument at every $z\ne 0$ .
$\log z = \ln|z| + i\arg z$ has one branch for every choice of interval used to pin down $\arg z$ .