Principal branch

The standard, by-convention choice of branch, usually cut along the negative real axis.

The principal branch of a multivalued function is the specific branch this course (and most others) treats as the default, chosen once and for all by convention rather than by any mathematical necessity. For the argument and logarithm, the principal branch is cut along the negative real axis, restricting outputs to angles in $(-\pi,\pi]$ ; other courses or contexts sometimes cut elsewhere, so 'the' principal branch is really 'the usual' one.

Examples

The principal branch of $\log z$ , written $\operatorname{Log} z$ , is defined by $\operatorname{Log} z = \ln|z|+i\operatorname{Arg} z$ with $\operatorname{Arg} z\in(-\pi,\pi]$ .
Evaluating a formula 'at its principal branch' means using the $(-\pi,\pi]$ -valued convention rather than any of the other equally valid branches.