Principal value

The one output a principal branch reports, singled out by convention from a multivalued function's full set of answers.

The principal value of a multivalued expression is the single output selected by whatever principal branch applies to it -- the value reported once a convention (such as $\operatorname{Arg}\in(-\pi,\pi]$ ) has picked exactly one of the infinitely many equally valid answers. The principal argument and the principal logarithm are the two standing examples, but the same idea extends to any expression built from a multivalued function, such as a complex power.

Examples

The principal value of $\log(-1)$ is $i\pi$ , coming from $\operatorname{Arg}(-1)=\pi$ , even though $i\pi+2k\pi i$ is a valid logarithm of $-1$ for every integer $k$ .
Unless a problem says otherwise, 'evaluate $z^i$ ' in this course means: report its principal value.