Editing Exponentiation (section)

====Principal value====
The [[principal value]] of the [[complex logarithm]] is the unique continuous function, commonly denoted <math>\log,</math> such that, for every nonzero complex number {{mvar|z}},
: <math>e^{\log z}=z,</math>
and the [[argument (complex analysis)|argument]] of {{mvar|z}} satisfies
: <math>-\pi <\operatorname{Arg}z \le \pi.</math>
The principal value of the complex logarithm is not defined for <math>z=0,</math> it is [[continuous function|discontinuous]] at negative real values of {{mvar|z}}, and it is [[holomorphic]] (that is, complex differentiable) elsewhere. If {{mvar|z}} is real and positive, the principal value of the complex logarithm is the natural logarithm: <math>\log z=\ln z.</math>

The principal value of <math>z^w</math> is defined as
<math>z^w=e^{w\log z},</math>
where <math>\log z</math> is the principal value of the logarithm.

The function <math>(z,w)\to z^w</math> is holomorphic except in the neighbourhood of the points where {{mvar|z}} is real and nonpositive.

If {{mvar|z}} is real and positive, the principal value of <math>z^w</math> equals its usual value defined above. If <math>w=1/n,</math> where {{mvar|n}} is an integer, this principal value is the same as the one defined above.