section_path: Non-integer powers of complex numbers/Complex exponentiation
section_name: Complex exponentiation
paragraph_text: 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.