section_path: Non-integer powers of complex numbers/Complex exponentiation/Multivalued function
section_name: Multivalued function
paragraph_text: Multivalued function
  
  In some contexts, there is a problem with the discontinuity of the principal values of log z {\displaystyle \log z} and z w {\displaystyle z^{w}} at the negative real values of z. In this case, it is useful to consider these functions as multivalued functions.
  
  If log z {\displaystyle \log z} denotes one of the values of the multivalued logarithm (typically its principal value), the other values are 2 i k π + log z, {\displaystyle 2ik\pi +\log z,} where k is any integer. Similarly, if z w {\displaystyle z^{w}} is one value of the exponentiation, then the other values are given by
  
  where k is any integer.
  
  Different values of k give different values of z w {\displaystyle z^{w}} unless w is a rational number, that is, there is an integer d such that dw is an integer. This results from the periodicity of the exponential function, more specifically, that e a = e b {\displaystyle e^{a}=e^{b}} if and only if a − b {\displaystyle a-b} is an integer multiple of 2 π i. {\displaystyle 2\pi i.}
  
  If w = m n {\displaystyle w={\frac {m}{n}}} is a rational number with m and n coprime integers with n > 0, {\displaystyle n>0,} then z w {\displaystyle z^{w}} has exactly n values. In the case m = 1, {\displaystyle m=1,} these values are the same as those described in § n th roots of a complex number. If w is an integer, there is only one value that agrees with that of § Integer exponents.
  
  The multivalued exponentiation is holomorphic for z ≠ 0, {\displaystyle z\neq 0,} in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. If z varies continuously along a circle around 0, then, after a turn, the value of z w {\displaystyle z^{w}} has changed of sheet.