Editing Exponentiation (section)

====Multivalued function====
In some contexts, there is a problem with the discontinuity of the principal values of <math>\log z</math> and <math>z^w</math> at the negative real values of {{mvar|z}}. In this case, it is useful to consider these functions as [[multivalued function]]s.

If <math>\log z</math> denotes one of the values of the multivalued logarithm (typically its principal value), the other values are <math>2ik\pi +\log z,</math> where {{mvar|k}} is any integer. Similarly, if <math>z^w</math> is one value of the exponentiation, then the other values are given by
: <math>e^{w(2ik\pi +\log z)} = z^we^{2ik\pi w},</math>
where {{mvar|k}} is any integer.

Different values of {{mvar|k}} give different values of <math>z^w</math> unless {{mvar|w}} is a [[rational number]], that is, there is an integer {{mvar|d}} such that {{mvar|dw}} is an integer. This results from the [[periodic function|periodicity]] of the exponential function, more specifically, that <math>e^a=e^b</math> if and only if <math>a-b</math> is an integer multiple of <math>2\pi i.</math>

If <math>w=\frac mn</math> is a rational number with {{mvar|m}} and {{mvar|n}} [[coprime integers]] with <math>n>0,</math> then <math>z^w</math> has exactly {{mvar|n}} values. In the case <math>m=1,</math> these values are the same as those described in [[#nth roots of a complex number|§ {{mvar|n}}th roots of a complex number]]. If {{mvar|w}} is an integer, there is only one value that agrees with that of {{slink||Integer exponents}}.

The multivalued exponentiation is holomorphic for <math>z\ne 0,</math> in the sense that its [[graph of a function|graph]] consists of several sheets that define each a holomorphic function in the neighborhood of every point. If {{mvar|z}} varies continuously along a circle around {{math|0}}, then, after a turn, the value of <math>z^w</math> has changed of sheet.