text: The multivalued exponentiation is holomorphic for  in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. If  varies continuously along a circle around , then, after a turn, the value of  has changed of sheet.
  Computation
  The canonical form  of  can be computed from the canonical form of  and . Although this can be described by a single formula, it is clearer to split the computation in several steps.
  Polar form of . If  is the canonical form of  ( and  being real), then its polar form is  where  and  (see atan2 for the definition of this function).
  Logarithm of . The principal value of this logarithm is  where  denotes the natural logarithm. The other values of the logarithm are obtained by adding  for any integer .
  Canonical form of  If  with  and  real, the values of  are  the principal value corresponding to
  Final result. Using the identities  and  one gets  with  for the principal value.
  Examples
  The polar form of  is  and the values of  are thus  It follows that So, all values of  are real, the principal one being
  Similarly, the polar form of  is  So, the above described method gives the values In this case, all the values have the same argument  and different absolute values.
  In both examples, all values of  have the same argument. More generally, this is true if and only if the real part of  is an integer.