stylistic:style_transfer, structural:reformatting
Complex exponentiation

Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for z w z^{w}. So, either a principal value is defined, which is not continuous for the values of z that are real and nonpositive, or z w z^{w} is defined as a multivalued function.

In all cases, the complex logarithm is used to define complex exponentiation as

where log z {\displaystyle \log z} is the variant of the complex logarithm that is used, which is, a function or a multivalued function such that

for every z in its domain of definition.