Multivalued function

A rule assigning more than one output to some input, like the complex argument or logarithm.

A multivalued function assigns more than one output value to at least one input, unlike an ordinary function, which by definition assigns exactly one. The complex argument and the complex logarithm are the two running examples in this course: every nonzero $z$ has infinitely many valid arguments (differing by multiples of $2\pi$ ) and infinitely many valid logarithms as a result. A branch is the standard way to tame a multivalued function back down into an ordinary, single-valued one on part of its domain.

Examples

The argument of $z=i$ is multivalued: $\pi/2$ , $\pi/2+2\pi$ , $\pi/2-2\pi$ , and every other $\pi/2+2k\pi$ are all equally valid.
Because $e^{i\theta}$ has period $2\pi$ , the equation $e^w=z$ has infinitely many solutions $w$ for any nonzero $z$ , making $\log z$ multivalued.