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 . So, either a principal value is defined, which is not continuous for the values of  that are real and nonpositive, or  is defined as a multivalued function.
  In all cases, the complex logarithm is used to define complex exponentiation as
  where  is the variant of the complex logarithm that is used, which is, a function or a multivalued function such that
  for every  in its domain of definition.
  Principal value
  The principal value of the complex logarithm is the unique continuous function, commonly denoted  such that, for every nonzero complex number ,
  and the argument of  satisfies
  The principal value of the complex logarithm is not defined for  it is discontinuous at negative real values of , and it is holomorphic (that is, complex differentiable) elsewhere. If  is real and positive, the principal value of the complex logarithm is the natural logarithm:
  The principal value of  is defined as
  where  is the principal value of the logarithm.
  The function  is holomorphic except in the neighbourhood of the points where  is real and nonpositive.